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Under mathematics come a number of different branches, of which one is algebra. You need to learn mathematics from all aspects to order to shine in your field of practice....
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Use this guide in helping one better understand the properties and rules within algebra, trigonometry, and statistics....
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1,138The theorems of the propositional calculus and the predicate calculus are stated, and the analogous principles of Boolean Algebra are identified. Also, the primary principles of modal logic are stated, and a procedure is described for identifying their Boolean analogues. | 677.169 | 1 |
About this product
Description
Description
DESCRIPTION The TI-84 Plus series graphing calculators is are the de facto standard for graphing calculators used by students in grades 6 through college. With so many features and functions, the TI-84 Plus graphing calculator can be a little intimidating. Using the TI-84 Plus is an easy-to-follow guide to using these calculators for class and for the SAT and ACT. It starts with a hands-on orientation to the calculator so readers will be comfortable with its menus, buttons, and the special vocabulary it uses. Then, it explores key features while tackling problems just like the ones seen in math and sciences classes. TI-84 Plus calculators are permitted on most standardized tests, so the book provides specific guidance for SAT and ACT math. Along the way, easy-to-find reference sidebars offer skills in a nutshell for those times when just a quick reminder is needed. KEY SELLING POINTS Includes coverage of the brand-new TI-84 Plus CE For TI-83 Plus and TI-84 Plus series of graphing calculators The missing manual for the TI-84 Plus calculators Gets readers up and running on calculators fast Fun, engaging, and approachable examples Easy hands-on learn by doing approach AUDIENCE This book is written for students, teachers-anyone who wants to use the TI-84 Plus or TI-83 Plus of graphing calculators. No prior experience is needed and it assumes advanced kwledge of math and science. ABOUT THE TECHNOLOGY The TI-84 Plus series is the de facto standard for graphing calculators used by students in grades 6 through college and for standardized tests. These calculators can do everything from basic arithmetic through graphing, pre-calculus, calculus, statistics, and probability, and are even great tools for learning programming.
Author Biography
Christopher Mitchell is a teacher, student, and recognized leader in the graphing calculator enthusiast community. You'll find Christopher (aka Kerm Martian) and his community of calculator experts answering questions, sharing advice, and providing educational tools on his website cemetech.net. | 677.169 | 1 |
These three worksheets combined have 180 quadratic equations.
On the first page of 'Solving Quadratic Equations 1' the equations are organised by type of factorising: Highest Common Factor, Difference of Two Squares and Product-Sum Method. On the second page, the same three types have been mixed up, and students need to identify which type of factorising to use.
'Solving Quadratic Equations 2' and 'Solving Quadratic Equations 3' are more difficult, and can be used as a differentiated task either done in class or set for homework. 'Solving Quadratic Equations 3' is targeting students who are aiming for a grade 9.
All three worksheets have answers.
For your convenience, the Word files are included.
10 pages of neatly handwritten notes on the whole syllabus of Pure Core 1 (AQA). I usually use these to give to my students if they struggle to take notes and concentrate at the same time. It also acts as a handy detailed syllabus/plan of study, as well as just a revision tool for students:
Sections of the notes:
1 - Straight Lines
2 - Quadratics
3 - Surds
4 - Simultaneous Equations
5 - Inequalities
6 - Algebraic Division
7 - Cubic Graphs
8 - Circles
9 - Differentiation
10 - Integration
* - With 2 pages of worked exam solutions to go with regular examples throughout.
Please have a look at some of my free IB notes if you would like to see the style of notes I am offering here. Hopefully there are none, but please let me know if you do see any errors! Also, let me know if there is any interest in other A-Level modules!
This worksheet bundle contains questions on binomial expansion for positive integer power index n.
Detailed typed answers are provided to every question. I hope you find it useful. You can get more worksheets on many topics, mix and match, with detailed step-by-step solutions at
Presentation covering the whole of chapter 4 (6-8hrs) from the Pearson edexcel Pure Mathematics Year 1/AS book. Includes examples and notes for:
Cubic graphs
Quartic graphs
Reciprocal graphs
Points of intersection
Graph transformations
Translating graphs
Stretching graphs
Transforming functions
Suitable for the new A-Level specification year 1 or AS-level. Also suitable for able GCSE students.
Presentation designed to be used as teaching aid for entire block of lessons. Opportunities for students to work through examples on whiteboards, and notes on key points. All learning objectives included and maintained throughout the presentation. Editable for you to select you own questions or add consolidation work.
OTHER CHAPTERS AVAILABLE
Aimed at KS5 pupils and pupils doing further maths IGCSE
full lesson with ppt and worksheets (worksheets are from SRWhitehouse thanks for posting them as the work set in the PPT relates to Edexcel IGCSE Further Pure Maths text book)
PPt has full worked examples, starter on finding the descriminant and finding how many roots a quadratic has). Introduces the roots and sums and products. Opportunity for group discussion with full work examples. (amended one slide 03/11)
This is a whole lesson. 20 slides.
It starts looking at drawing cubic graphs, before quickly moving on to looking at sketching them and developing the understanding and skills to do so. It looks at working out the key points needed and using the factor theorem and factorising in order to do so. The lesson comes with a starer, several little MWB activities, worksheet, excellent teaching slides, a handout of all the different graphs and plenary.
NOTE: Feel free to browse my shop for more excellent free and premium resources and as always please rate and feedback, thank you.
This is lesson on remainder and factor theorems.
The bundle contains both the lesson presentations with 'overlay animations', where steps are reviewed one at a time. There is also an accompanying presenter view pdf for teachers for each lesson ppt. The documents are prepared meticulously with LaTeX, I hope you find these useful.
In addition, you can get unlimited number of high quality practice questions and solutions for both lessons on my website: They are under topics "Polynomial Division". You can generate the worksheets yourself, and each question will come with a set of very detailed step by step solutions, set out in exactly the same way as the lesson presentation examples.
Hope you find these useful, and if you have any comments or suggestions, please feel free to contact me through TES or my website.
Best wishes,
Joe
This is a three lesson sequence of lessons on quadratic graphs, with the first two containing the material needed for GCSE and the third lesson the additional material needed for AS-Level.
1. Introduction
2. Gradients and Other Problems
3. Further Problems
This is second of two whole lessons on teaching the various aspects of quadratic graphs containing the various aspects introduced with the new 9-1 GCSE syllabus. The first lesson being the introduction is important to go through before this lesson and this builds on it by looking at problems involving the gradient. 14 slides.
This lesson is available with two other lessons on quadratic graphs as a three lesson bundle and a reduction and is highly recommended as an excellent sequence of lessons.
NOTE: Feel free to browse my shop for more excellent free and premium resources, and as always please rate and feedback, thank you. | 677.169 | 1 |
Course Description: In Math 3400, we will discuss prime numbers
and the
Fundamental Theorem of Arithmetic. Congruences play a crucial role in
this
course. We will cover several special congruences such as Wilson's
Theorem,
Fermat's Little Theorem, and Euler's Theorem. In addition, we will
investigate
the Perpetual Calendar and, as a fun application of congruences, we
will learn
how to quickly compute the weekday of any given date! We will also
cover
various other important topics in Number Theory such as Multiplicative
Functions, Primitive Roots, and Quadratic Residues.
Homework:Here
is
a
complete list of homework problems for this course. Homework will be
assigned
at the end of each lecture. Homework will be due on Fridays by 11:50 am
and you
can turn it in before or after (but not during) class. Here
you can
check to see which sections are due when. Homework will be returned in class. Late homework will NOT be accepted for any
reason.
Your lowest two homework scores will be dropped.
Exams &
Grading
Policy: Your final grade
will be
based on your homework (your lowest two homework scores will be
dropped), two
midterms, and a comprehensive final. The midterms will be on February
24 and
April 14. The final exam will be on May 7th. Please make sure that you
are
available at those dates, since Number
Theory. You are expected to come to lecture on time. Plan ahead so you
are not
late. You should come to every lecture, and come prepared. If you have
to miss
class or if you are late for some unavoidable reason it is your
responsibility Number Theory from watching the professor or friends display
ideas and
solve problems. You must try problems, finish problems, ask questions,
correct
your mistakes, put concepts in your own words, and practice, practice,
practice! The good news: A small increase in effort usually results in
a big
increase in success!
Disabilities: It is the responsibility of students with
certified
disabilities to provide the instructor with appropriate documentation
from the Dean
of Students Office.
Cheating: No cheating will be tolerated. Anyone
caught cheating will receive an F
in the course. Furthermore, a letter will be sent to the appropriate
dean.
Lecture schedule: In the very unlikely event that you missed
a lecture
you can check here to see what material I covered in class. I will
update the
Actual Lecture Schedule after every class.
Actual
Lecture Schedule
Mo
Jan
13
1.1-1.3
We
Jan
15
1.4+3.1
Fr
Jan
17
3.1-3.2
We
Jan
22
3.2-3.4
Fr
Jan
24
HW
Mo
Jan
27
3.4
We
Jan
29
3.4-3.5
Fr
Jan
31
3.5+HW
Mo
Feb
03
3.6+4.1
We
Feb
05
4.1
Fr
Feb
07
4.1+HW
Mo
Feb
10
4.2-4.3
We
Feb
12
5.1
Fr
Feb
14
HW
Mo
Feb
17
5.2
We
Feb
19
6.1
Fr
Feb
21
Review
Mo
Feb
24
Midterm
We
Feb
26
UNT closed
No class
Fr
Feb
28
6.1
Mo
Mar
03
6.2-6.3
We
Mar
05
6.3+7.1
Fr
Mar
07
HW
Mo
Mar
10
7.1
We
Mar
12
7.1
Fr
Mar
14
HW
Mo
Mar
24
7.2-7.3
We
Mar
26
7.3
Fr
Mar
28
7.4
Mo
Mar
31
9.1
We
Apr
02
9.1
Fr
Apr
04
HW
Mo
Apr
07
9.2
We
Apr
09
9.2-9.3
Fr
Apr
11
Review
Mo
Apr
14
Midterm
We
Apr
16
9.3
Fr
Apr
18
9.3-9.4
Mo
Apr
21
9.4
We
Apr
23
11.1
Fr
Apr
25
HW
Mo
Apr
28
11.1
We
Apr
30
11.2
Fr
May
02
Review
We
May
07
FINAL 10:30-12:30pm
The following
Tentative
Lecture Schedule gives you an idea what material I intend to cover in
this
class, but NOTE that the Actual Lecture Schedule (above) might be
different! | 677.169 | 1 |
MATH Pre Calcul Advice
Showing 1 to 3 of 4
This class provided me with preparation for a very vigorous class which I am yet to take- AP Calculus.
Course highlights:
This course helped me recap my previous courses like algebra 1 and 2 with a twist of trigenometry, all in an effort to provide insight for calculus, which is yet to come.
Hours per week:
3-5 hours
Advice for students:
I would advise students to pay close attention even though you may be familiar with the topic because this is all being re-taught in an effort to make sure you truly understood the material to avoid any confusion in the future when your taking Calculus.
Course Term:Fall 2017
Professor:kasmaii
Course Tags:Background Knowledge ExpectedGreat Intro to the SubjectMany Small Assignments
Dec 26, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
I recommend taking this course because it is a great opportunity to start with a course that introduces you to the course you will be taking the year after. It will get you prepared with skills and the goal of the course.
Course highlights:
From this course I gained knowledge and expectations in what would you be learning in Calculus AB. In the course I learned new formulas that would be used in most of calculus and new techniques to solve a problem.
Hours per week:
6-8 hours
Advice for students:
In order to succeed in this class is to dedicate time and study each lesson you learn. Each new lesson you learn it will be another step from a previous lesson. If you don't understand a lesson don't hesitate to ask for help it will help you achieve. Don't study a day before a test study various days before in case you have a new question of a doubt and if you learn better in groups, create study groups!
Course Term:Fall 2015
Professor:gomez
Course Required?Yes
Course Tags:Great Intro to the SubjectMany Small AssignmentsA Few Big Assignments
Nov 07, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
The teacher is a very good at teaching her students.
Course highlights:
Different components and equations need for calculus.
Hours per week:
6-8 hours
Advice for students:
Be prepared to study and memorize equations.
Course Term:Fall 2015
Professor:Nava
Course Tags:Great Intro to the SubjectLots of WritingMany Small Assignments | 677.169 | 1 |
TRIGONOMETRY, 2E, International Edition is designed to help you learn to "think mathematically." With this text, you can stop merely memorizing facts and mimicking examples-and instead develop true, lasting problem-solving skills. Clear and easy to read, TRIGONOMETRY, 2E, International Edition illustrates how trigonometry is used and applied in the real world, and helps you understand how it can apply to your own life.
About the Author Lothar Redlin grew up on Vancouver Island, received a Bachelor of Science degree from the University of Victoria, and a Ph.D. from McMaster University in 1978. He subsequently did research and taught at the University of Washington, the University of Waterloo, and California State University, Long Beach. He is currently Professor of Mathematics at The Pennsylvania State University, Abington Campus. His research field is topology. Saleem Watson received his Bachelor of Science degree from Andrews University in Michigan. He did graduate studies at Dalhousie University and McMaster University, where he received his Ph.D. in 1978. He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland. He also taught at The Pennsylvania State University. He is currently Professor of Mathematics at California State University, Long Beach. His research field is functional analysis.
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Unformatted text preview: Sets and Functions Sets and Functions If there is one unifying foundation common to all branches of mathematics, it is the theory of sets. Definition A set is a collection of objects characterized by some defining property. The objects in a set are called elements or members of the set. We use capital letters to designate sets, lowercase letters to designate elements and the symbol 2 to denote membership in a set, and a = 2 B means a is not an element of B . MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 28 Sets and Functions Sets and Functions For example, MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 28 Sets and Functions Sets and Functions For example, If A = f 1 ; 2 ; 3 ; 4 g , then 2 2 A and 5 = 2 A . Said another way, we require that the sentence " a 2 A " be a statement: MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 28 Sets and Functions Sets and Functions For example, If A = f 1 ; 2 ; 3 ; 4 g , then 2 2 A and 5 = 2 A ....
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Full Document | 677.169 | 1 |
In a few courses, all it's going to take to move an examination is take note using, memorization, and remember. Even so, exceeding within a math class can take a unique variety of energy. You cannot simply exhibit up to get a lecture and watch your teacher "talk" about calculus and . You discover it by executing: paying attention at school, actively studying, and solving math issues – even though your instructor hasn't assigned you any. For those who find yourself having difficulties to carry out properly inside your math course, then stop by very best website for solving math troubles to learn the way you may become a greater math student.
Low-cost math authorities online
Math courses comply with a pure progression – each builds on the awareness you've attained and mastered from the previous program. In case you are finding it tough to adhere to new ideas in class, pull out your previous math notes and assessment prior material to refresh yourself. Ensure that you fulfill the stipulations before signing up for the course.
Assessment Notes The Evening In advance of Course
Dislike whenever a trainer calls on you and you have overlooked ways to resolve a specific difficulty? Keep away from this moment by reviewing your math notes. This may assist you to identify which ideas or queries you'd prefer to go over in school another working day.
The considered doing research each and every night may seem bothersome, but if you want to succeed in , it is important that you constantly practice and learn the problem-solving methods. Use your textbook or on the internet guides to operate via prime math difficulties on a weekly basis – regardless if you have no homework assigned.
Use the Health supplements That include Your Textbook
Textbook publishers have enriched fashionable publications with further product (like CD-ROMs or on line modules) which can be used to help students get additional exercise in . A few of these elements could also incorporate a solution or clarification guideline, which might enable you to with doing work as a result of math difficulties all on your own.
Examine Ahead To stay In advance
If you need to minimize your in-class workload or maybe the time you spend on research, use your free time just after college or on the weekends to read forward for the chapters and ideas that will be included the subsequent time you happen to be at school.
Evaluate Old Assessments and Classroom Illustrations
The operate you are doing in school, for homework, and on quizzes can offer clues to what your midterm or closing examination will search like. Use your old exams and classwork to make a personal research manual to your upcoming examination. Appear at the way your teacher frames questions – this is certainly possibly how they may surface on your own examination.
Figure out how to Operate Via the Clock
This is a preferred study suggestion for people taking timed tests; specifically standardized checks. For those who have only forty minutes for just a 100-point exam, you'll be able to optimally expend four minutes on every 10-point problem. Get info regarding how prolonged the check are going to be and which kinds of inquiries will likely be on it. Then prepare to assault the simpler concerns 1st, leaving on your own enough time and energy to commit about the extra hard types.
Maximize your Assets to get math homework enable
If you are owning a hard time knowing principles in class, then be sure to get aid beyond class. Check with your mates to produce a analyze group and visit your instructor's business office hours to go over tricky difficulties one-on-one. Show up at review and evaluation sessions whenever your teacher announces them, or use a private tutor if you want one.
Converse To Yourself
Whenever you are reviewing difficulties for an test, try out to explain out loud what strategy and methods you used to get your methods. These verbal declarations will occur in useful through a test whenever you really need to remember the steps you need to just take to locate a resolution. Get more follow by making an attempt this tactic by using a pal.
Use Examine Guides For Added Apply
Are your textbook or course notes not serving to you have an understanding of whatever you need to be finding out in school? Use analyze guides for standardized tests, including the ACT, SAT, or DSST, to brush up on outdated content, or . Review guides generally appear outfitted with complete explanations of the way to resolve a sample trouble, and you simply can normally locate exactly where will be the improved acquire mathchallenges. | 677.169 | 1 |
Podcast Intro to Linear Functions
5 April 2013
Create a brief (3–5 min) audio podcast. In the podcast, I want you to reveal either one of your essential questions or your unit questions. The audio file will become an introduction to your learning module, intended to pique student curiosity and lead them into inquiry of the subject matter.
These are the podcast intros to my created lesson in graphing linear functions in algebra. I used Garage Band for the editing and placed them at the Internet Archive. Students would be instructed to listen to part 1, perform a mental exercise, and then continue on with part 2. | 677.169 | 1 |
Information
A bit of information concerning fanlistings in general and calculusCalculus
Calculus (from Latin, "pebble" or "little stone") is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. Calculus has widespread applications in science and engineering and is used to solve complex and expansive problems for which algebra alone is insufficient. It builds on analytic geometry and mathematical analysis and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus. | 677.169 | 1 |
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Express review guides. Basic math and pre-algebra.
Thorough introduction prepares students to take basic Algebra examinations. Includes pretest that pinpoints stengths and weaknesses providing guidance in areas where students need to focus their attention. A posttest provides detailed answer explanations and proof of improvement.
Abstract:
Thorough introduction prepares students to take basic Algebra examinations. Includes pretest that pinpoints stengths and weaknesses providing guidance in areas where students need to focus their attention. A posttest provides detailed answer explanations and proof of improvement. | 677.169 | 1 |
We have had Elementary Algebra by Harold Jacobs floating around our homeschool since my oldest child was in 7th grade. It has been used as a reference, a supplement to another curriculum, as a review, and is now being used as the main Algebra I curriculum for my 9th graders.
Although this text is out of print, it is widely used and can often be found on websites that sell used books and curriculum. There are also companion books available such as a complete solutions manual, a teacher's guide, and test banks. We have never used it, but there is also a DVD course that uses this text through "Ask Dr. Callahan."
For the purpose of this review, I will focus on using Elementary Algebra as our main Algebra I curriculum.
Jacobs presents the lessons in a conversational style, addressing the student directly. Each lesson starts out with a "real world" example and leads the student through several worked problems. The lessons are presented in a discovery method, with the student learning more about the topic as they work through the exercises in a set.
The topics on the book progress and build upon the previous topics, so grasping a concept is necessary before moving on. There are plenty of opportunities to practice new concepts in each problem set, and the Test Bank also offers additional problems and multiple tests for each chapter.
This text is word problem heavy, often building on the topics within the word problems themselves in a discovery style.
My 9th graders prefer to do their work independently, only coming to me when they do not understand a lesson or are struggling with a concept. They have been able to work through most of this text in this manner, but do occasionally run into an issue of needing a teacher to fill in some gaps in order for them to grasp the concept. This text was written for classroom use, and this is very evident in some lessons.
There is a teacher's guide that gives additional lesson material, but personally it hasn't been useful to me. I often use other sources to flesh out a lesson if I am having trouble explaining it to my students.
I feel like this curriculum is working very well with my students who do not struggle with math in general. This text couldn't have been used independently with my older child as it does not mesh with her learning style at all, and she needs more direct instruction. If a parent is very comfortable with teaching Algebra, then this could be a good fit for a student not as strong in math when taught one on one.
We began publishing this collaborative blog in 2013, but the concept and vision go much farther back. We've been talking about it for years! We've been rewarded for our hard work in so many ways: We've had the fun of knowing what it's like for friends to grow into a team centered around a cause. We've had the chance to hone our mission and consider our opinions, and look back over our own experiences as classical home educators. We've learned that we have hidden talents and strengths, and a passion for sharing our love of learning.
We will cherish this award from the homeschooling community. We're proud to be here, and excited to meet new friends in the blogosphere as we gear up for a wonderful year of blogging together here at Sandbox to Socrates in 2014. Thanks again, Homeschool Blog Awards!
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"Enlighten people generally, and tyranny and oppressions of body and mind will vanish like evil spirits at the dawn of day." -Thomas Jefferson
The United States of America is a country in which we purport to hold education of the young as a most treasured value. We work to hard to educate our own population. During the 2009-2010 school year, federal, state and local governments in the US spent over $638 billion dollars on elementary and secondary schools [1]. We have risked the lives of our soldiers to build schools in Afghanistan. Prior to the fall of the Taliban, only 32 percent of Afghanistan's school aged children were enrolled in school–only three percent of girls. The US worked to build and refurbish hundreds of schools, resulting in millions of children (including a large percentage of girls and young women) being allowed to enroll in school [2].
Another important principle dearly held is the lack of government censorship in the US. In fact, we sanction other governments when they impose censorship upon their people. Recently, the US imposed sanctions upon Iran for engaging in satellite jamming and limiting access to the internet by their populace. Victoria Nuland, spokesperson for the US Department of State, said in her press release dated 8 November, "Countless activists, journalists, lawyers, students, and artists have been detained, censured, tortured, or forcibly prevented from exercising their human rights. With the measures we are taking today, we draw the world's attention to the scope of the regime's insidious actions, which oppress its own people and violate Iran's own laws and international obligations. We will continue to stand with the Iranian people in their quest to protect their dignity and freedoms and prevent the Iranian Government from creating an "electronic curtain" to cut Iranian citizens off from the rest of the world." [3]
Americans generally hold the view that education is always a positive. Therefore, one would think that Massive Open Online Courses (MOOCs) would be viewed as a boon to our civilization and a great benefit of technology to the modern age. Coursera is one such provider of MOOCs to students around the globe. It came as a surprise to many when the US sanctions intended to punish the government of Iran included the blocking of Coursera [4] and other MOOCs to Iran. We are going to punish the government of Iran for blocking access to internet information from its people by blocking internet educational information from its people? On what planet does this make sense?
If you are an American, please urge your government officials to exempt MOOCs from government sanctions upon Syria, Iran, Cuba and other countries in which a free, expansive alternative educational system is advantageous to a populace that otherwise hears only government ideology in the vacuum that exists when the free exchange of ideas is taken away. Education in this case should not be considered a commodity to be blocked from the people of Iran or any other sanctioned government, but be considered valued knowledge and information which will benefit the global community.
Contact the US State Department's Office of Foreign Assets Control here:
Jen. Jen has two more children who are equally smart, but learned to read on a more average schedule.
There have been a few blog posts around the internet lately on a phrase that Andrew Kern is famous for, "teaching from a state of rest." It's one that has left many a homeschooling mother scratching her head for hours; frankly, I've only been able to understand it as a few ideas have come colliding together in my own heart. Though there are many soft and grace-filled posts on the state of rest out there, this is not one that is soft. Grace is favor, the free and undeserved help God gives us to respond to his call to become children of God, adopted sons, and partakers of the divine nature and of eternal life. Sometimes grace comes in the form of a clue-by-four: this post is for those who need a little more definition in how this works out in our lives, people like myself.
Other blogs have wonderful posts on this idea and how to attain it, but I'm going to come at it from another angle: that rest starts with observing our unmet expectations and what those expectations mean, and what they shine a spotlight on. I've written before on Sandbox to Socrates about homeschooling being a spotlight on what can be wrong in our households; in this case, homeschooling can be a spotlight on what is wrong in our hearts.
There is a great homily on Audio Sancto called Sloth: the Vice of Homeschoolers. When I didn't understand the meaning of the word sloth, I was pretty taken aback by that title.
sloth
noun ˈslȯth, ˈsläth also ˈslōth
: the quality or state of being lazy
: a type of animal that lives in trees in South and Central America and that moves very slowly
Sloth is often summed up as laziness, but a truer definition is not doing what we are supposed to be doing, when we are supposed to be doing it. The cure for being slothful is knowing our place (you will hear more on that in the audio homily), which is doing what you are supposed to be doing, when you are supposed to be doing it.
Meaning, if your house is spotless, but the children's education has fallen off the pier, that is sloth. If you have been running around like a chicken with her head off, but you are supposed to be resting, you are being slothful. If you are bound up in unmet expectations of your child's education, are buying heaps of curriculum in hopes that it will be THE thing that gets them into Harvard, if your heart is anxious (when you are supposed to be resting in trust) you are being slothful. If you are piling worksheet after worksheet in front of your child because more of any work = success, you might be slothful.
Sometimes when sloth doesn't look like laziness, it is shining a spotlight on our idols. What makes us anxious? Impatient? Angry? Bitter? Most of the time, it is unmet expectations. Unmet expectations of what? That our children would be gifted students and they are 'only' average? That the work would be easier? That our days would look like some fictionalized ideal in our heads? That the monotony of the day wouldn't make us think that if we were out there, with a career or job, we would be doing something useful with our lives? That someone, anyone, would be a better teacher than we are?
Do we have more pride in our teaching ability, rather than trusting our children's needs being met through us? Are we anxious and fearful of doing the wrong thing because everyone else is doing something different? Comparisons lead us to constantly question ourselves and the paths our families are on. Are we questioning our vocations as mothers and homeschoolers because the outside world looks prettier and more rewarding when our egos are are bruised because we're 'just' stay at home mothers?
Look, sometimes we DO need to just clean up the house, and get the meals on the table, and put our noses to the grindstone, to pull ourselves up by the bootstraps because we've fallen. But even then, there is rest. There is rest because of trust. There is trust in the calling, in the vocation in our lives that is marriage and the upbringing of our children; trust in the love of Christ because he will not lead us astray; trust that when we DO get off track, He writes straight with crooked lines.
In the end, with sloth it all comes down to ego, to what we think what should be–and wasn't pride the first sin? Us forgetting who God is and our place. In Him.
So. How do we teach from a state of rest? By repenting. Sometimes when we think of repenting, we think of sackcloth and ashes. But that's not what it is. It means to turn around, to change your mind. Doesn't that sound much easier–to walk toward something better? But part of repenting is acknowledging that we need to change our minds. Let's not be so stuck in our ways that we are unable to change our minds.
"'The beginning of salvation is to condemn oneself' (Evagrius). Repentance marks the starting-point of our journey. The Greek term metanoia…signifies primarily a 'change of mind.' Correctly understood, repentance is not negative but positive. It means not self-pity or remorse but conversion, the re-centering of our whole life upon the Trinity. It is to look not backward with regret but forward with home–not downwards at our own shortcomings but upwards at God's love. It is to see, not what we have failed to be, but what by divine grace we can now become; and it is to act upon what we see. To repent is to open our eyes to the light. In this sense, repentance is not just a single act, an initial step, but a continuing state, an attitude of the heart and will that needs to be ceaselessly renewed up to the end of life. In the words of St Isaias of Sketis, 'God requires us to go on repenting until our last breath.' 'This life has been given you for repentance," says St Isaac the Syrian. 'Do not waste it on other things.'" Met. Kallistos WareAge 6, grade Prep
This picture really doesn't do the experience justice. Imagine standing on a hill just before dawn and seeing shooting stars, that fade as the sun rises and brightens the sky…
Age 6, grade Prep.
Age 6, grade Prep
Remember that brightly coloured waterfall? This time it is blue.
Age 6, grade PrepAge 6, grade Prep
Rose-Marie was one of those enthusiastic planners who began researching when she was pregnant with her first. She wanted to homeschool because it sounded like an affordable adventure, then she met her kids personally…DD is 6 years old and has Echolalia and some processing issues so isn't speaking fluently yet. DS is 4 years old, has retained primitive reflexes and while there may be a deity somewhere who knows what's going to happen with this kid, he/she/it hasn't chosen to inform us. They live on a hill in rural southern Australia without enough solar panels and like it there.
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When I first thought about homeschooling, I envisioned happy children, eagerly learning from the vast array of materials I would lovingly provide for them. Crafts, field trips, music lessons, our world of home education would be glorious. They'd excel at everything, all veritable geniuses and perfectly behaved children, and my home would be filled with the sounds of children's laughter, the smell of home-made meals, and always neat before my husband came home.
I don't know whose children I thought I'd be homeschooling, but it evidently wasn't mine. I also seemed to believe that homeschooling would cause me to have a complete personality change.
Needless to say, I'm the same woman I've always been, just with the added responsibility of educating my children.
If my inherent personality flaws weren't enough, then LIFE had to happen.
Winds of change have gusted through our life, some wonderful (new babies) some ill (chronic pain disability, job loss). I've had people ask, half in awe, half in horror, how I can possibly keep homeschooling through it.
How? I don't know. I just do. I have what I call "Dory days," those days when all you can do is keep swimming. Not that you seem to be actually getting ahead in any way, shape or form, but motion, even if it's not seeming to get you closer to your goal, is better than complete inertia.
The fact that I'm more stubborn than the average definitely helps. The fact that my children have inherited this trait is a mixed bag. It depends on if they're with me, or against me. Tazzie, our soon to be nine-year old, deciding that he can't read, doesn't like reading, doesn't want to read, absolutely has been working against me, but I'm more stubborn than he is. I think.
One thing I've absolutely had to do is pack up my idea of what 'should be' and deal with 'what is'. A curricula that I thought would be wonderful just didn't work for my kids. My careful plans were shredded by chaos. I've had to learn flexibility, patience, and to quit daydreaming of dumping my kids on the steps of the local public school, yelling, "They're yours now!" and fleeing, cackling wildly.
For me, it's the small accomplishments that keep me going. For example, Princess, our seven-year old, finally *getting* how this whole phonics gig works gets me through trying to teach Tazzie long division. His constant cries of, "I don't geeeeeet iiiiittttttt!" and the wash, rinse, repeat, of showing him, yet again, how it works. If it wasn't for the bright moments of what I think of as "the Click," I'm honestly not sure how I would manage, but I suspect it wouldn't be in completely socially acceptable ways. I know they'll get there, eventually, though. After all, the Grand Canyon started out as a river over some rocks, right?
I have some good friends who just awe me with their homeschooling. Honestly, they intimidate me too. I think we all have those folks in our lives, who from the outside seem to have it altogether. They're who we want to be when we grow up: we want to parent like them, teach like them, keep house like them.
The reality of that is, they'd probably be completely horrified by the idea.
And that right there, folks, is the true secret of homeschooling. There's not a single one of us that's convinced we're doing this all right. There's not a homeschooling family that hasn't made some compromises along the way, who've had to identify what their priorities really are, and let other things slide. We can't do it all, and we need to be honest with ourselves about that. We need to give ourselves grace. This is especially important when challenges hit. And challenges *will* hit, no matter who you are. They may be huge, such as relocating, job loss, health issues. They may be small, such as folks wandering around in bathing suits in the winter because laundry hasn't been done in recent memory, or having toast and cereal for supper for the third time in a week because all available kitchen work space is consumed with science and art projects.
Having a sense of humour, giving yourself grace, and being patient and kind to yourself are survival skills when it comes to homeschooling. These are also valuable life skills to give your children, giving them a cornerstone for their future that's as necessary as math and reading.
(This article is published by StS with permission from Melissa Charles) Melissa, more commonly known as 'Mom' or 'Imp', hails from Canada. She spends her day Wife-ing to 'Wolf', and Mom-ing 'Diva', 'Tazzie', 'Princess', 'Toddler Terror/Boo' and the newest addition to the minion roster, 'Cubby'.
When not home educating, attempting to control the chaos that is every day life, and dodging bears and deer, she can be found blogging one armed at Not A Stepford Life
This semester marked a milestone for our family: all five of our children have either graduated or are currently in college. It's been an interesting journey.
We started thirteen years ago, when our oldest daughter was a sophomore in high school. We had no idea what we were doing, and neither did the college. We decided it was wise to begin with a class where our student knew most of the material, in order to learn "classroom". We hit on "Fundamentals of Music," basic music theory. Then it got complicated. We started with SAT/ACT scores (which were high enough for admittance to the college) and a homemade transcript, and I trudged up the stairs to the registrar on the third floor. I was told she needed instructor permission as she was underage. I tracked down the professor – not too hard as he was her orchestra director – and then trudged back up the stairs. Then I was told that we needed this and that, and I swear, I lost ten pounds on those stairs! The final straw was being told, "The school won't pay for it; she's not a junior." I whipped out my checkbook and exclaimed, "HER school will pay for it. Do you want this money or not? If not, I'll be speaking to the college president." The tune changed dramatically. We finally got her accepted and registered, and the college figured out how to put her in the computer.
All was good: she had the highest grade in the class; and she learned how to deal with folks wanting to borrow notes and other students trying to cheat off of her. She went on to graduate high school with over 31 credits, all of which transferred when she began her undergraduate degree at Hillsdale College. Hillsdale even helped design her senior year to insure credit transference. It allowed her to graduate with Honors in four years.
We learned something new with daughter #2: to be cautious with the college as, unexpectedly, they matriculated her (i.e. they declared her a high school graduate and a degree-seeking student). This made her ineligible for high school sports and could also have fouled up NCAA eligibility. We learned to check EACH year that they hadn't graduated her, again. We ran into another unforeseen question: if she was a full-time college student, would she be ineligible for high school sports? We could not get a response in writing from the high school athletics association, so she dropped a class. We also navigated placement tests with this daughter, as she wanted to take math classes. She took some music theory, several science courses, battled some calculus, and took a bunny trail of several architectural classes.
She added something new to the mix: several overseas classes from Hillsdale College. The course on WWII was a life-changer. She was standing under the Eiffel Tower on the 4th of July when she recognized Sgt. Malarkey of "Band of Brothers" fame. That chance encounter set her on a new path, one that lead to appointments at the Naval Academy, the Air Force Academy, and the Coast Guard Academy. She chose to sing "Anchors Aweigh" and now flies helicopters for the US Navy. None of her dual enrollment credits transferred, as the academies don't accept outside credits, but we were told her 45 DE credits were what got her accepted.
Our middle child's interests did not lie in academic pursuits, but in more hands-on experiences. With that in mind, she began her college career with Lifeguard Training. Again, we picked a course where she could excel as she was a strong swimmer. It was a tough first semester, with 3 hours of swimming for lifeguarding, coupled with 2 hours a day of high school swim team, along with an hour per day of diving. We could smell the chlorine on her from across the room. She did some academic classes, such as science, writing, and programming, but she much preferred Emergency Response and Firefighter I. Again, we kept a close eye on matriculation status. She graduated with 29 credits, most of which transferred to the University of Wyoming.
Our son has thrived in the college setting, taking such things as academic writing, three physics classes, four programming classes, three math classes, Emergency Response, and the ever-popular Lifeguard Training. He will be just under full-time status this final semester. Again, the university matriculated him. With a change in how high school students are registered, we are now paying far less per semester hour than previously. That's a relief, as he'll graduate with 47 credits. We found that having a wide range of professors to write recommendation letters is a very good thing. We purposely chose our son's courses so as to have English, science, and math teachers. He also took the WWII class from Hillsdale College, and again, it was a profound experience. He wrote one of his college application essays on his thoughts while standing in the American cemetery at Normandy Beach.
This brings us to our youngest. She's begun her college career at 14 with Music Fundamentals, earning the highest grade in the class. She'll tackle Lifeguard Training next semester, keeping up the tradition of reeking of chlorine. Next fall, she'll jump into computer applications. From there we'll see where her interests lie.
The kids have learned a wide range of subjects and have been taught by some leading experts in their fields. It's exciting to hear my son come home jazzed about the presentation from his physics professor or the latest cool trick from his computer science professor. They've learned time management skills, group dynamics, deadlines, and organization. As parents, we've learned to have early ACT/SATs, to have strong up-to-date transcripts, to pick the first few professors carefully, and to keep on top of the registrar. We've learned to be flexible with home school courses during college midterms and finals. I've been there to show the kids how to navigate a college bookstore, explained the importance of keeping the syllabus, and what prerequisites mean.
Dual enrollment has been a very successful part of our homeschool journey, and we're grateful that our children have had a chance to experience it.
Heart Cross Ranch is the mom of five children, three of whom have graduated. She is in her 26th year of homeschooling, with just three left to go! She lives high up in the Colorado mountains, in the nation's icebox, on a cattle and sheep ranch. She enjoys being heavily involved with Boy Scouts, taking sports photos for the local paper, and anything chocolate. She confesses that much of her "homeschooling" consists of throwing interesting books at her children.
Our own Jane-Emily is guest-blogging today over at To the Moon and Back! As a librarian, Jane-Emily definitely knows the best ways to introduce our early elementary-aged children to the skills and knowledge they need as they learn to see the library as a home away from home. Please drop in at To the Moon and Back to read Jane-Emily's suggestions!
Thanks again to our good friend Dusty for hosting us today on herlovely blog where she discusses all things pertaining to homeschooling, marriage, parenting, and faith.
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It is not easy to me to explain about hatred and intolerance to my children. They have lead a somewhat sheltered life; the idea of treating people differently because of how they look is a foreign concept to them and one that I would prefer not to teach them about. But you cannot teach bravery without teaching about fear; you can not teach about Martin Luther King Jr. without explaining injustice.
Our lesson plan for this MLK Jr. Day includes reading the books Happy Birthday, Martin Luther King by Jean Marzollo and Young Martin Luther King, Jr. "I Have a Dream" , a Troll First-Start Biography by Joanne Mattern. The BrainPOP video for Marin Luther King, Jr. is free to watch, (please pre-watch, especially if you have younger children) and you can watch the footage of King's "I Have a Dream" speech here, below. It's almost 17 minutes long, so while my kids are listening to it I'll have them coloring a picture I got from this website that will encourage the kids to look for other skin-tone crayons to color with instead of just "peach."
For a sweet activity to explain how even though we look different on the outside, we're the same on the inside, we will snack on M & M candies. First we'll bite them in half to see how while the outside shell is different colors, they're all chocolate inside. Then I'll put an M & M in their mouths and ask if they can tell what color it is just by the taste. We'll do that several times to make the point that they all taste the same.
Older relatives are a great resource for first hand experiences from this time period. We'll be interviewing Grandpa over the phone to add to the lesson as well.
I don't know how or if teaching about this will change how my children look at the world. They are only 7 and 4 1/2. But I hope that this will set a good foundation for further learning when they get older.
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The Beginning
This Fall, I'll be an incoming freshman taking my school's equivalence to "Intro. to Proofs" class. However, I've been wanting to go ahead and start already, so I picked up Spivak's Calculus book, as many people suggested it to me as "real" Math. So far, the content itself is easy and understandable. His approach at the Delta-epsilon proofs was rather beautiful, if I do say so myself. However, when it comes to the problems, I just collapse. I'm able to some, but nowhere near enough to say "I've done enough for this section." It worries me. Am I not intelligent enough to be a mathematician? I mean, some of these proofs are doable, but then some I'm not able to understand them even after reading the answer manual. Granted, I'm not used spending so much time on problems (let's be honest, I used Stewart's for my Calculus knowledge and we all know those problems weren't exactly brain-busters) and perhaps I'm not spending adequate on time on them, but I can't help but feel like I should able to do them if they were in the problem set.
Should I be worrying that I'm struggling so? Is Spivak too much for someone who hasn't even taken their first proof class? How should one begin one's "mathematical enlightenment" for lack of better words? Am I just wasting my time? Or am I just worrying for nothing?
First of all, I think you are prematurely judging whether or not you are fit to be a mathematician. Spivak's exercises, while difficult, do not necessarily reflect how well you can handle more advanced math courses. Learning the concepts well is more important than being able to come up with the trick that will easily lead you to a solution to a particular problem.
Having said that, if you want to become a better problem-solver, you have to spend more time thinking about the exercises. If you can't figure out the gist of a problem immediately, then the problem deserves more thought. You should not ever read an entire solution in the solution manual, but only enough to maybe provide you with the right first step. On the other hand, the problem may be that you haven't understood the material in the chapter well enough. Yes, Spivak is fairly easy to digest, but if you can't motivate every step of the proof on your own, then you won't be ready for most of the problems.
I think you should reread the chapters in Spivak and see if you can reproduce the proof on your own. Don't worry too much about the formal write-up, but make sure you can fill in the logical steps. Then see if the problems become clearer.
Something I've noticed with many proof-heavy math texts is that they like to stick real ballbuster problems after each chapter.
I'm pretty sure in a few of these cases, the theorem took months or years to prove originally. You, however, are expected to solve it in under a week, because you have all the necessary tools supplied to you in the preceding chapter.
Don't get too discouraged, though! As long as you can knock a few of them out, you're still in the clear. Being able to do every. single. problem. in a math text is a madman's pursuit. (It just happens that there are a fair number of less-than-sane mathematicians).
Make an honest attempt at each problem you can. After you work with a subject enough, you still might not be able to prove the theorem, but you can provide an outline for how it probably needs to be proved. You make medium-to-large leaps in logic whenever you get stuck, with confidence that a more careful or harder working person could go back and work out the annoying details.
This proves that if 3 divides [itex]k^2[/itex] then 3 also divides [itex]k[/itex].
Here's another way of looking at this: You know that [itex]3[/itex] is a prime number, so no product of primes unequal to [itex]3[/itex] will ever equal [itex]3[/itex]. Now suppose that [itex]3[/itex] divides [itex]k^2[/itex] but not [itex]k[/itex]. Do you see that you instantly run into a contradiction? This is a reason why that statement is simple.
if sqrt(a) is rational then
sqrt(a)=b/c a fraction in simplest form (c = 1 is the only time it is rational)
a = b^2/c^2
if a is an integer, there must be a common factor in b^2 and c^2 meaning that b/c could not have been in simplest form.
proof by contradiction.
does that work?
if a = 3, it holds've seen "iff" but I only it means "if and only if" and I have no idea what the difference between that and just if means. xD.
I haven't seen any logical connectives or anything of the sort. I'm taking the equivalent of intro to proofs (called Sequences, Series and Foundations) this fall which, judging from the course descriptions, covers everything you mentioned.
And yeah...delta-epsilon...oh god, haha. I can do only the basic cases. Perhaps you're right. Either way, I'm just hoping to get some good stuff out of Spivak and perhaps be fully prepared for the class in the fall.
I'd recommend picking up a copy of How to Prove It: A Structured Approach by Daniel J. Velleman.
It nicely introduces all of the logical and set theoretic foundations one needs to explore different proof techniques and then uses these techniques to work out problems from function theory, number theory, relations, and cardinality. Nice book.
Don't worry about the [itex]\delta , \varepsilon [/itex] proofs; everyone struggles with these. Fortunately only a few are needed to bootstrap more sophisticated methods in the lower level courses. By the time they become more commonplace, hopefully one has had more exposure and experience with proof techniques.
Specifically, using the epsilon/delta definition in proofs demands that you have a VERY solid grasp on the quantification of the variables and the exact meaning of the logical connectiveI've seen "iff" but I only it means "if and only if" and I have no idea what the difference between that and just if means. xD.
"If" is a logical implication. It's only one way, though! If a number is divisible by 6, then it's even. That's a true statement. If you "turn around" an if statement, it's not necessarily true. "If a number is even, then it's divisible by 6" clearly isn't.
If you want to "flip" an if statement around, the best you can do is negate both sides. This is called the contrapositive. It's the basis of proof by contradiction. "If a number ISN'T even, then it ISN'T divisible by 6".
If an if statement really DOES work both ways, the two statements are logically equivalent and we sometimes say "if and only if" or "iff". "A number is congruent to 0 mod 2 iff it's even". You can flip these around all you want onceThere are still subtleties to what the qualifiers mean and how they work. For the epsilon-delta, keep in mind that since the delta quantifier exists inside the scope of epsilon's scope, delta may be defined in terms of epsilon. Another way to say it is epsilon is a constant with respect to deltaDon't sweat it, man. I was in the same boat you were, and I'm actually probably not that far ahead of you. I'm an applied mathematics major at an engineering school. Translation: my intro calculus classes had NO theory (the engineers would likely have revolted if there was some).
You're self-studying Spivak, which means you have curiosity and initiative. Those are VERY important traits. Don't sell yourself short.
Do you know who Terence Tao is? If you were to look up "child-prodigy" in the dictionary, his picture would be right there. He's now a prof at UCLA...read this: second Union68. I am a practicing engineer, with a lot of experience doing analytical
"analysis" (in a loose sense of the word). The past year or so I have been trying to self-learn some more theoretical math, mainly just for fun, but partially so that my eyes don't glaze over when I read some of the more high-brow engineering literature.
Anyway, a lot of books were too intimidating for me, and I had to struggle too much and would give up very quickly. Reading a book that explicitly taught about logic, quantifiers, relations, sets, and structures of proofs has helped a bunch! The other folks here know better than I what books to recommend on such topics; I found one that worked for me but it is uninspiring. I have made a lot of progress, and you can too! If you take the proofs class next semester and work hard at it, I would bet that by this time next year you will be amazed at your progress, and Spivak will be more fun!
Also, remember that your profs and TAs have been struggling with the material for years, and that is how we all learn. They are not necessarily a lot smarter or more gifted than you, just because it looks like this stuff is easy for them. I know that when I TA'd upper division electrodynamics I had taken at least 5 courses at the same or higher level - yes the problems for that class were pretty easy for me, but only because I had spent hundreds of hours solving electrodynamics problems and struggling with the material on my own. It was HARD for me the first time through, that is for sure!
You have great initiative - more than I did before I started school. You can do this!I have not read Spivak, so I cannot comment on its utility, but I do instruct from Stewart's (our college is shifting from Essential Calculus to the sixth edition Calculus) and I have found it servicable, not scintilating, but functional and complete enough to expose freshmen to the fundamentals of differential and integral calculus.
I have always found that texts that focus on exercises that can only be solved using "tricks" or clever solution methods tend not to prepare students for the reality that many times things just don't behave in a friendly or elegant way and brute force is the appropriate approach.
However, a strong enough exposure to the theory behind how things work (and just as importantly under what conditions it doesn't) can be invaluable in adapting techniques to new situations.
Not to demean anyone, but I have run into more than a few students who've been shown the "plug-and-play" recipe methods for too long who get lost when faced with something they haven't encountered yet. I suspect a little theory can help in a pinch.
The other thing I encounter is the impression that the proofs or derivations should somehow be obvious. Not so. It took many mathematicians often decades to solve some of the theorems that are sometimes offhandedly summarized beginning with "obviously ..." The impossibility of cubing the circle took more than a dozen centuries!!!
So if you aren't getting things at first, do not despair; many luminaries wracked their brains trying to solve these things too (many of then unsuccessfully). We are learning from their trials - whether successful or not.
I have found that true understanding of how everything works comes gradually, some gaps being filled in only after subsequent work causes things to fall into place. Trying to explain it to someone else is often a telling test of how well one grasps something as one will often be asked unexpected questions. | 677.169 | 1 |
The How and Why of One Variable Calculus
First course calculus texts have traditionally been either
"engineering/science-oriented" with too little rigor,
or have thrown students in the deep end with a rigorous analysis
text. The How and Why of One Variable Calculus closes this
gap in providing a rigorous treatment that takes an original and
valuable approach between calculus and analysis. Logically
organized and also very clear and user-friendly, it covers 6 main
topics; real numbers, sequences, continuity, differentiation,
integration, and series. It is primarily concerned with developing
an understanding of the tools of calculus. The author presents
numerous examples and exercises that illustrate how the techniques
of calculus have universal application.
The How and Why of One Variable Calculus presents an
excellent text for a first course in calculus for students in the
mathematical sciences, statistics and analytics, as well as a text
for a bridge course between single and multi-variable calculus as
well as between single variable calculus and upper level theory
courses for math majors | 677.169 | 1 |
9780072429770
0072429771105.09
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Summary
Maintaining its hallmark features of carefully detailed explanations and accessible pedagogy, this edition also addresses the AMATYC and NCTM Standards. In addition to the changes incorporated into the text, a new integrated video series and multimedia tutorial program are also available. Designed for a one-semester basic math course, this successful worktext is appropriate for lecture, learning center, laboratory, or self-paced courses.
Table of Contents
Preface
xiii
To the Student
xxv
Operations on Whole Numbers
1
(126)
Pre-Test
2
(1)
The Decimal Place-Value System
3
(8)
Addition
11
(18)
Using a Scientific Calculator to Add
25
(4)
Subtraction
29
(14)
Using a Scientific Calculator to Subtract
41
(2)
Rounding, Estimation, and Order
43
(10)
Multiplication
53
(18)
Division
71
(20)
Using a Scientific Calculator to Divide
87
(4)
Exponential Notation and the Order of Operations
91
(12)
Using a Scientific Calculator to Evaluate an Expression
99
(4)
Solving Geometric Applications
103
(24)
Summary
117
(4)
Summary Exercises
121
(4)
Self-Test
125
(2)
Multiplying and Dividing Fractions
127
(96)
Pre-Test
128
(1)
Prime Numbers and Divisibility
129
(8)
Factoring Whole Numbers
137
(10)
Fraction Basics
147
(12)
Simplifying Fractions
159
(10)
Using Your Calculator to Simplify Fraction
167
(2)
Multiplying Fractions
169
(10)
Using Your Calculator to Multiply Fractions
177
(2)
Applications of Multiplication of Fractions
179
(8)
Dividing Fractions
187
(14)
Using Your Calculator to Divide Fractions
199
(2)
*Computer-Related Applications: Time
201
(22)
Summary
209
(4)
Summary Exercises
213
(4)
Self-Test
217
(2)
Cumulative Test
219
(4)
Adding and Subtracting Fractions
223
(66)
Pre-Test
224
(1)
Adding and Subtracting Fractions with Like Denominators
225
(8)
Common Multiples
233
(10)
Adding and Subtracting Fractions with Unlike Denominators
243
(18)
Using Your Calculator to Add and Subtract Fractions
257
(4)
Adding and Subtracting Mixed Numbers
261
(12)
Using Your Calculator to Add and Subtract Mixed Numbers
271
(2)
Estimation Applications
273
(16)
Summary
279
(2)
Summary Exercises
281
(4)
Self-Test
285
(2)
Cumulative Test
287
(2)
Decimals
289
(122)
Pre-Test
290
(1)
Place Value and Rounding
291
(12)
Adding and Subtracting Decimals
303
(22)
Using Your Calculator to Add or Subtract Decimals
321
(4)
Multiplying Decimals
325
(16)
Using Your Calculator to Multiply Decimals
335
(6)
Area and Circumference
341
(14)
Dividing Decimals
355
(16)
Using Your Calculator to Divide Decimals
369
(2)
Converting from Fractions to Decimals
371
(8)
Converting from Decimals to Fractions
379
(8)
Using Your Calculator to Convert Between Decimals and Fractions
383
(4)
Square Roots and the Pythagorean Theorem
387
(24)
Using Your Calculator to Find Square Roots
397
(4)
Summary
401
(2)
Summary Exercises
403
(4)
Self-Test
407
(2)
Cumulative Test
409
(2)
Ratios and Proportions
411
(58)
Pre-Test
412
(1)
Ratios
413
(8)
Rates and Unit Pricing
421
(8)
Proportions
429
(8)
Solving Proportions
437
(6)
Solving Applications of Proportions
443
(26)
Using Your Calculator to Solve Proportions
453
(6)
Summary
459
(2)
Summary Exercises
461
(4)
Self-Test
465
(2)
Cumulative Test
467
(2)
Percents
469
(66)
Pre-Test
470
(1)
Changing a Percent to a Fraction or a Decimal
471
(10)
Changing a Decimal or Fraction to a Percent
481
(8)
Identifying Rate, Base, and Amount
489
(6)
Three Types of Percent Problems
495
(8)
Solving Percent Applications
503
(32)
Using Your Calculator to Solve Percent Problems
519
(6)
Summary
525
(2)
Summary Exercises
527
(4)
Self-Test
531
(2)
Cumulative Test
533
(2)
Geometry and Measure
535
(78)
Pre-Test
536
(3)
The Units of the English System
539
(12)
Metric Units of Length
551
(10)
Metric Units of Weight and Volume
561
(14)
Using Your Calculator to Convert Between the English and Metric Systems | 677.169 | 1 |
Physics Formulas and Equations sheet
Physics and mathematics is all about formulas and equations. There would be rarely any question that would be solved without any equation or a formula behind your answer. Every chapter has its own specific formula and many formulas have been derived from one an other and concepts usually grow and continue.
Here is a list of formulas that would help students from get their formulas. This formula sheet would work for all students whether it would be a Cambridge GCE O Level / A Level student, Edexcel GCSE / IGCSE/ A Level, AQA, WJEC and 10, 11, 12 Level students can study from it with ease. These formulas can be used both at secondary and higher level to solve questions from past papers and worksheets.
By practice you would be able to learn these formulas and you needto memorize and they should be in our brain while reading the question so you plan the solution already in your mind. In this way you would be able to master all formulas. | 677.169 | 1 |
Pre-requisites
Restrictions
Overview
The concept of symmetry is one of the most fruitful ideas through which mankind has tried to understand order and beauty in nature and art. This module first develops the concept of symmetry in geometry. It subsequently discusses links with the fundamental notion of a group in algebra. Outline syllabus includes: Groups from geometry; Permutations; Basic group theory; Action of groups and applications to (i) isometries of regular polyhedra; (ii) counting colouring problems; Matrix groups.
Learning outcomes
The intended subject specific learning outcomes.
On successfully completing the module students will be able to:
1 demonstrate knowledge and critical understanding of the well-established principles within basic group theory and symmetries;
2 demonstrate the capability to use a range of established techniques and a reasonable level of skill in calculation and manipulation of the material to solve problems in the following areas: isometries of the plane, groups, action of groups, matrix groups, symmetric groups, cyclic groups and dihedral groups;
3 apply the concepts and principles in group theory in well-defined contexts beyond those in which they were first studied, showing the ability to evaluate critically the appropriateness of different tools and techniques.
The intended generic learning outcomes.
On successfully completing the module students will be able to:
Demonstrate an increased ability to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation.
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 |
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Unformatted text preview: MT 3802 NUMERICAL ANALYSIS 2008/2009 Dr Clare E Parnell and Dr St´ ephane R´ egnier October 1, 2008 Chapter 0 Handout 0.1 Notation Throughout this course we will be using scalars, vectors and matrices. It is essential that you know what they are and can tell the difference between them!! • Scalar: e.g. α , β or γ . Scalars belong to a field F such as R or C . • Vectors: e.g. x or f ( x ). The first vector belongs to a Vector Space (defined later) such as R n , C n and is of the form x = ( x i , x 2 , . . . , x n ). The second vector is a continuous function such as the polynomial x 2 + x and belongs to a vector space such as C (-∞ , ∞ ). In the lectures, vectors will be denoted by: x- i.e. a small letter with a wiggly line under and in the online lecture notes by x- i.e. a bold small letter . • Matrices: e.g. A or B . An n × n matrix has the form A = a 11 a 12 . . . a 1 n a 21 a 22 . . . . . . . . . . . . . . . . . . a n 1 . . . . . . a nn . In the lectures, a matrix will be denoted by: A- a capital letter with a straight line under 1 and in the online lecture notes by...
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I'm just wondering if anyone can give me a few pointers here so that I can understand the basics of free 8th grade math worksheets. I find solving equations really difficult. I work in the evening and thus have no time left to take extra tutoring . Can you guys suggest any online tool that can assist me with this subject?
I have a good recommendation that could help you with math . You simply need a good program to make clear the problems that are hard . You don't need a tutor , because on one hand it's very expensive , and on the other hand you won't have it near you whenever you need help. A software is better because you only have to get it once, and it's yours forever . I advice you to try Algebrator, because it's the best. Since it can resolve almost any math problems , you will certainly use it for a very long time, just like I did. I got it years ago when I was in Pre Algebra, but I still use it sometimes .
That's true, a good software can do miracles . I used a few but Algebrator is the greatest. It doesn't matter what class you are in, I myself used it in Basic Math and Intermediate algebra too, so you don't have to worry that it's not on your level. If you never used a software before I can tell you it's very easy , you don't have to know much about the computer to use it. You just have to type in the keywords of the exercise, and then the program solves it step by step, so you get more than just the answer.
I remember having often faced problems with quadratic inequalities, matrices and distance of points. A truly great piece of algebra program is Algebrator software. By simply typing in a problem homework a step by step solution would appear by a click on Solve. I have used it through many algebra classes – Intermediate algebra, Algebra 2 and Intermediate algebra. I greatly recommend the program.
Thank you, I will try the suggested program . I have never tried any software until now, I didn't even know that they exist. But it sure sounds amazing ! Where did you find the program? I want to purchase it right away, so I have time to get ready for the test . | 677.169 | 1 |
Numerical Linear Algebra for Applications in Statistics(Paperback)
Synopsis
Accurate and efficient computer algorithms for factoring matrices, solving linear systems of equations, and extracting eigenvalues and eigenvectors. Regardless of the software system used, the book describes and gives examples of the use of modern computer software for numerical linear algebra. It begins with a discussion of the basics of numerical computations, and then describes the relevant properties of matrix inverses, factorisations, matrix and vector norms, and other topics in linear algebra. The book is essentially self- contained, with the topics addressed constituting the essential material for an introductory course in statistical computing. Numerous exercises allow the text to be used for a first course in statistical computing or as supplementary text for various courses that emphasise comput | 677.169 | 1 |
College Courses
Mathematics
MATH 101 Introduction to Finite Mathematics (3)
This course emphasizes problem solving and critical thinking as it introduces students to basic concepts in arithmetic, symbolic logic, number theory, set theory, elementary probability, and statistics.
MATH 110 Financial Mathematics (3)
This mathematics course reviews the fundamentals of algebra and financial applications. Concepts of linear systems
are applied to time value equations including simple and compound interest. Geometric progressions are used to
study simple and general annuities, equations of value, amortization, sinking funds, and bonds. Students are
encouraged to see the relevance of mathematical concepts in their lives and in the business world and develop
specific math skills that are useful in many areas of life.
MATH 292 Quantitative Methods (3)
Topics covered in this course include probability, decision analysis, sampling distributions, applications for
sampling and risk analysis, statistical estimation and hypothesis testing, regression and correlation, analysis of
variance, time series and index numbers, and an introduction to linear programming. Microcomputer software is
used to illustrate statistical concepts | 677.169 | 1 |
MATH1220: Further Reading
1. Peter R. Cromwell, Polyhedra, Cambridge University Press 1997. A very
accessible book on Polyhedra.
2. C.G. Gibson, Elementary Geometry of Differentiable Curves, Cambridge University
Press 2001. A hands-on introduction to the study of curves using
calculus.
3. Kai Hauser and Reinhard Lang, On the geometrical and physical meaning of Newton's Solution to Kepler's problem, Mathematical Intelligencer, Vol. 25 no. 4, p35--44.
This is a very recent article in a very good mathematics journal.
Many of the articles are quite hard for undergraduates but this one (also bit hard!) is worth a look. You will find this journal on Level 8 of the library.
4. Leonard Mlodinow, Euclid's window : the story of geometry from parallel lines to hyperspace, Allen Lane 2002. A popular science book on geometry which
gets into physics at the end. | 677.169 | 1 |
This best-selling book provides an accessible introduction to discrete mathematics, using an algorithmic approach that focuses on problem-solving techniques. The new edition weaves techniques of proofs into the text as a running theme. Each chapter has a special section dedicated to showing students how to attack and solve problems.
About the Author:
Richard Johnsonbaugh is Professor Emeritus of Computer Science, Telecommunications and Information Systems, DePaul University, Chicago. Prior to his 20-year service at DePaul University, he was a member and sometime chair of the mathematics departments at Morehouse College and Chicago State University. He has a B.A. degree in mathematics from Yale University, M.A. and Ph.D. degrees in mathematics from the University of Oregon, and an M.S. degree in computer science from the University of Illinois, Chicago. His most recent research interests are in pattern recognition, programming languages, algorithms, and discrete mathematics. He is the author or co-author of numerous books and articles in these areas. Several of his books have been translated into various languages. He is a member of the Mathematical Association of America. | 677.169 | 1 |
Mathematics
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This bestselling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, providing students with a solid foundation in the principles of mathematical thinking. Comprehensive and evenly paced, the book provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help students develop insight into mathematical ideas. The authors' attention to detail and clarity—the same as found in James Stewart's market-leading Calculus text—is what makes this text the proven market leader. | 677.169 | 1 |
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Forex for Ambitious Beginners is a guide to successful currency trading. It will help you avoid
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Dwarfs your fear towards complicated mathematical derivations and proofs. Experience Kalman filter with hands-on examples
to grasp the essence.A book long awaited by anyone who could not dare to put their first step into Kalman filter. The author presents Kalman ... | 677.169 | 1 |
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Fundamental Algebra and Trigonometry
Similarly, whenever you make any mistake while solving sums, you also note that down in your diary. But don't take our word for it, go to Time and Date: Do you like electronics, amateur radio, etc? This is a summary of the results in the above pages, for the derivation of these formulae see the links above. And, for the final project the teacher, the student, and classmates will evaluate using a rubric. Maybe I'll rewrite them over here so we have them on the board.
Pages: 0
Publisher: Addison-Wesley Pub. Co. (1977)
ISBN: 020103767X
Complete trigonometry - Primary Source Edition
Mathematical Tables of Elementary and Some Higher Mathemataical Functions Including Trigonometric Functions of Decimals of Degrees of Logarithms
An Elementary Treatise on the Application of Trigonometry to Orthographic and Stereographic Projection Dialling Mensuration of Heights and Distances ... of the University at Cambridge, New England.
College Algebra and Trigonometry
Trigonometry. Part 2: Algebraical Trigonometry.
If you are looking for a program to help improve your multiplication skills, this just might be the app for you download! However, as the semester wears on, the students are increasingly replaced with tape recorders until eventually even the teacher is replaced by a reel-to-reel recording, leaving our hero to brave the terrors of calculus alone Trigonometry with calculators. Let's start by stating some (hopefully) obvious limits: Since each of the above functions is continuous at x = 0, the value of the limit at x = 0 is the value of the function at x = 0; this follows from the definition of limits A modern approach to algebra and trigonometry (International textbooks in mathematics). Rational expressions online calculator, softmath.com, multiplying rational expressions calculator, mcdougal littell answers, logarithm and exponential function linear and quadratic equation, permutation combinations sketchpad Algebra & Trigonometry with Analytic Geometry, Math 025, Indiana University. Prove the Laws of Sines and Cosines and use them to solve problems. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces) Easy Outline of Precalculus. Finding the answer to this question were mathematicians such as Ptolemy and Al-Khwarizmi of Baghdad (ca. 790–840). Interestingly, Al-Khwarizmi's name is the root for the word "algorithm" and whose book, al-jabr, is the root for the word algebra. The early triangle trigonometry problems led to the formation of trigonometric identities download Fundamental Algebra and Trigonometry pdf. You will need the email address of your friend or family member. Proceed with the checkout process as usual. Once you have paid for your order, your friend or loved one will receive an email letting them know that they have a gift waiting for them at TheGreatCourses.com Manual of Trigonometry. It provides demonstrations and grades problems. The disadvantage of software is that you have to sit in front of the computer: not good for more than one kid at a time. Advanced Math software is available from a number of publishers, including Grasp Math ( ). DIVE Advanced Math software from Genesis Science ( ) works through all the problems in the Saxon textbook Title: Holt Algebra with Trigonometry 2.
The Mandelbrot set is an example of a fractal ( See ) A Modern Course in Trigonometry. There is a high amount of academic rigor stressed in the community, in particular, the parents of many of these �advanced� students. There are three other junior high level schools in Laramie, but the Laramie Junior High School has more junior high students than the other three schools combined Trigonometry (Instructor's Annotated Edition) by John W. Coburn (2008) Hardcover. Dividing Scientific Notation technical math, subtracting polynomials calculator, grade 1 math trivia, aptitude questions and answers with explanation. Solve by grouping factor calculator, abstract sample of an investigatory project, fifth grade integer problems solving, mathematics for dummies +software, mcdougal littell geometry answers textbook answers, mcdougal littell grade 9 activation codes Natural Trigonometric Functions to Seven Decimal Places for Every Ten Seconds of Arc.
Trigonometry [With CDROM]
Once you have entered the size, you need to press the "Begin" button. Using your mouse, drag the shapes onto the box. When you can complete the square you win. A trigonometric identity is an equation involving trigonometric ratios of an angle, where the equation holds true for a defined range of values of the angle. For the right triangle ABC, let 0°≤ A ≤ 90° 1) cos2 A + sin2 A = 1. 2) cos2 A =1 - sin2 A. 3) sin2 A =1 - cos2 A. 4) sec2 A - tan2 A = 1. 5) 1 + tan2 A = sec2 A. 6) tan2 A = sec2 A – 1. 7) cosec2 A - cot2 A = 1. 8) cot2 A + 1 = cosec2 A The Element of Geometry. Practice addition, subtraction, multiplication, and division in an arcade game format. Blast the correct answer out of four before they hit the ground. This is an iPhone app, but will work on the iPAD. Mad Math is perfect for kids to practice their addition, subtraction, multiplication, and Division Facts read Fundamental Algebra and Trigonometry online. 13.75 mb Editors: Breu?, M., Bruckstein, A., Maragos, P., Wuhrer, S. (Eds.) Presentation of recent advances in the field of shape analysis Writt... 88.9 MB Algebra can be like a foreign language. But one text delivers an interpretation you can fully understand Plane Trigonometry and Complex Numbers. This AP Prep course will delve deeper into trigonometry while my unit is giving an overview and some particulars that apply directly to previous units HANDBUCH DER EBENEN UND SPHARISCHEN TRIGONOMETRIE. Students entering this sequence are to have mastered appropriate precalculus material and, in many cases, have had some previous experience with calculus in high school or elsewhere Outline of the Method of Conducting a Trigonometrical Survey, for the Formation of Geographical and. Hence our word "sine." [illustration source: George Gheverghese Joseph, The Crest of the Peacock: The Non-European Roots of Mathematics, new ed. (Princeton, NJ: Princeton University Press, 1991, 2000), p. 282.] Another set of trigonometric functions, tangent and cotangent, developed from the study of the lengths of shadows cast by objects of various heights Plane Trigonometry: Partial Solutions Manual.
A Descriptiv List of Books for the Young
Trig Graphs (Trigonometry Revision Book 4)
Algebra & Trigonometry with Applications
Plane and Spherical Trigonometry; An Elementary Text-Book
Elements of Geometry and Trigonometry with Applications in Mensuration.
Surveying and Navigation, With a Preliminary Treatise on Trigonometry and Mensuration
A collection of mathematical rules and tables; including interest, equation of payment, mensuration, geometry, and trigonometry, rules for calculating ... the circle: together with the metric system
Plane Trigonometry and Tables
College Algebra & Trigonometry
The merchant Leonardo Fibonacci of Pisa, who had learned about Arabic numerals in Tunis, wrote a treatise rejecting the abacus in favor of the Arab method of reckoning, and as a result, the system of Hindu-Arabic numeration caught on quickly in Central Italy New Plane and Spherical Trigonometry, Surveying and Navigation, Teachers' Edition - Primary Source Edition. All standards from the Common Core are addressed, with special considerations to the PARCC MCF and SBAC. Material is presented in a lesson format with follow-up interactive practice problems. I have two trigonometric problems that I solved, however it does not match the answer in the book: 1) A yacht crosses the start line of a race on a bearing of $31$ degrees Key To The Elements Of Plane Trigonometry: Containing Demonstrative Solutions Of All The Exercises In That Treatise (1862). A water surface is also a horizontal surface. The following definition will present no difficulty to the student. A plane surface is such that the straight line which joins any two points on it lies wholly in the surface. A plane surface is determined uniquely, by (a) Three points not in the same straight line, (b) Two intersecting straight lines Algebra and Trigonometry with Analytic Geometry. In order to see the steps, sign up for Mathway. Scroll through the topics to find the type of problem you want to check or practice. This will provide an example in the calculator so that you can see how it is formatted. You can then change the numbers or variables to fit the problem you are trying to check.[/box] [/frame]Parenthesis – They indicate multiplication or that the operation inside should be done first. [/frame]Brackets – Use brackets if you need a parenthesis within parenthesis – The brackets go on the outside as seen in this example: [3 + 2(10 -1)] ÷ 7. [/frame]Absolute Value – The absolute value tells how far away a number is from zero Test Bank for Swokowski and Cole's Algebra & Trigonometry With Analytic Geometry. Online math mcdougal littell book answers, absolute values in square and cube roots, QUADRATIC EQUATION BY COMPLETING THE SQUARE EXAMPLE, examples of quadratic expressions when prime, Create an expression for your classmates to solve that uses scientific notation and at least one of the rules for exponents online. How can you practice for the trigonometry placement test? The following topics may be covered on the placement test: Given the side lengths of a triangle, find the sin, cos, tan, cot, sec, and/or csc for a designated angle of the triangle The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh and Twelfth : The Errors, by Which Theon, Or Others, Have Long Ago ... Are Restored : Also, the Book of Euclid. Find the lengths of the sides and hypotenuse of the triangle. In a right triangle ABC with angle A equal to 90o, find angle B and C so that sin(B) = cos(B) Plane trigonometry together with logarithmic and trigonometric tables. Highly detailed, realistic appearance on all devices. Takes full advantage of the high resolution display ability of the iPhone 4 Scroll through operations to add, edit or delete entries in your calculation with realtime results like a spreadsheet. • Rotate the iPhone/iTouch/iPad to instantly swap between 4 different calculators. • Annotate a series of operations to turn your formula into a powerful spreadsheet style form. • Save formulas or operation history in your own math library. • Date Math is automatically detected in normal operations Plane and spherical trigonometry,: With tables. JMAP resources for the CCSS include Regents Exams in various formats, Regents Books sorting exam questions by CCSS: Topic, Date, Type and at Random, Regents Worksheets sorting exam questions by Type and at Random, an Algebra I Study Guide, and Algebra I Lesson Plans. Additional worksheets aligned to the CCSS are being developed for Algebra I, Geometry, and Algebra II download. | 677.169 | 1 |
Geometry of Curves and Surfaces with MAPLE
This concise text on geometry with computer modeling presents some elementary methods for analytical modeling and visualization of curves and surfaces. The author systematically examines such powerful tools as 2-D and 3-D animation of geometric images, transformations, shadows, and colors, and then further studies more complex problems in differential geometry. Well-illustrated with more than 350 figures---reproducible using Maple programs in the book---the work is devoted to three main areas: curves, surfaces, and polyhedra. Pedagogical benefits can be found in the large number of Maple programs, some of which are analogous to C++ programs, including those for splines and fractals. To avoid tedious typing, readers will be able to download many of the programs from the Birkhauser web site. Aimed at a broad audience of students, instructors of mathematics, computer scientists, and engineers who have knowledge of analytical geometry, i.e., method of coordinates, this text will be an excellent classroom resource or self-study reference. With over 100 stimulating exercises, problems and solutions, {\it Geometry of Curves and Surfaces with Maple} will integrate traditional differential and non- Euclidean geometries with more current computer algebra systems in a practical and user-friendly format.
"synopsis" may belong to another edition of this title.
Review:
"I was hunting for a book that would provide a set of practical exercises for the students of a graduate course entitled 'Geometric Modeling for Computer Graphics'.... The title of [this] book sounds appealing for such a purpose.... Almost every topic you could imagine about curves and surfaces is somewhere inside: this includes common, and less common, definitions and properties (parametric and implicit form, rectangular and polar form, tangent, asymptote, envelope, normal, curvature, torsion, twist, length, center of mass, evolute and involute, pedal and podoid, etc) as well as the whole menagerie of usual, and less usual, curves and surfaces (polynomials and rational polynomials, B-splines, Bezier, Hermite, Catmul--Rom, Beta-splines, scalar and vector fields, polygons and polyhedra, fractals, etc).
Of course 310 pages is a bit short to present all these topics deeply, but for each of them, there is at least a definition, an example, a piece of Maple source code and the resulting figure generated by the code (note that all the code pieces can be downloaded from the author's web page).... The index is rich enough to easily find a topic you are interested in.
To conclude, the book is clearly valuable for at least three kinds of people: first, people who are familiar with the mathematical aspect of curves and surfaces but unfamiliar with the computation and plotting possibilities providing by Maple; second, people who are familiar with Maple but unfamiliar with curves and surfaces; third, people who are unfamiliar with both topics." ― Computer Graphics Forum
"The book can be recommended to students of mathematics, engineering or computer science, who have already a basic knowledge of MAPLE and are interested in the visualizations of geometry." ---Zentralblatt MATH10533 | 677.169 | 1 |
Number and Quantity Algebra 1 Lessons The Real Number System N-RN Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from
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John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra | 677.169 | 1 |
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Piecewise Function Notes and Exercise
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This 6-page document contains a brief introduction to graphing piecewise function and writing linear functions for each segment of a piecewise function. It is appropriate to use in a classroom where teachers can guide students to plotting piecewise function using a table of value and properties of linear function.
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Required Materials:
Two (2) hardbound
composition notebooks (Mead). One for homework and other for
Journal Reflection. Only purchase notebooks with lines, no
gridded/graphing paper.
Texas Instruments
(TI) graphing calculator. This class will support TI-83, TI-84
(all in series), TI-nSpire (all in series). Other graphing
calculators may be used, but will not be supported during class
time. *If purchasing a calculator is a financial challenge,
please contact Mr. Germanis to make arrangements for a loan
calculator.
Daily Schedule
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unit lesson plan that was implemented to teach students
about Trigonometric functions. The following was displayed
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Mathematics students and teachers.
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As a teacher of undergraduate mathematics I found this book to be a valuable resource ... the material is presented in an uncluttered clear style that will enable undergraduates to use the book for reinforcement and/or alternative viewpoints on the mathematics they are learning. The exercises for the reader are about at the right level for an undergraduate and the mathematics covered is not just pure mathematics but essential topics in mathematical physics and probability theory.
Book Description World Scientific Publishing Co Pte Ltd. Paperback. Book Condition: new. BRAND NEW, A Mathematical Bridge: An Intuitive Journey in Higher Mathematics, Stephen Fletcher Hewson, An alternative introduction to the highlights of a typical undergraduate mathematics course, this book starts from very simple levels or principles and develops in an intuitive and entertaining way. The aim is to motivate and inspire the reader to discover and understand some of the truly amazing mathematical structures and ideas which are often swamped, or usually pass unnoticed or are not fully grasped, in a higher mathematics course. Bookseller Inventory # B9789812385550
Book Description World Scientific Publishing Co Pte Ltd, Singapore, 2003. Paperback. Book Condition: New. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. BTE9789812385550 | 677.169 | 1 |
How to Prepare Mathematics for SSC CGL Exam
How to Prepare Mathematics for SSC CGL Exam
Mathematics is arguably the most crucial part of the SSC CGL exam. It has the weightage of 50/200 in Prelims and 200/400 in Mains in SSC CGL. Quite often we have seen people scoring a perfect 200 and 50 in the Mathematics section. It is indisputably the crucial part of the exam that you cannot afford to ignore. Today, we will describe in this post on how to prepare for this important section in detail. Before going any further, let us first analyze this section in detail:
Mathematics in SSC CGL section
The level of SSC CGL Recruitment Exam has improved a lot in last 3-4 years. If you glance at the question papers of last 5 years, you'll see how the level has improved. SSC now has started giving more importance to the Advance Maths sections and the difficulty level has also increased. It is therefore quite essential to know the changing pattern, analyze it properly and put in your concentrated efforts accordingly.
Important Areas to Focus Upon
To improve your score in Mathematics, you need to improve your calculation speed and accuracy. You should remember tables, square, cubes and square roots for doing fast calculations.
More than 50% of the Mathematics paper including Prelims and Mains revolves around 4 major topics of the Advance Maths.
Algebra
A lot of questions are being asked from this topic. Some of the topics are Algebraic Identities, Factorization & Simplification of Polynomials, Simplification of Fractions and Age Determination. You should remember all the formulas and identities and then practice questions from this topic.
Geometry
This is the second most important topic and a lot of questions in SSC CGL 2016 were asked from this topic. Moreover, when combined with mensuration, it has a lot of weightage. SSC in 2016 has asked questions which involved a lot of calculations and there were very few questions that were direct or formula based. You need to practice questions from all chapters and also from previous year papers.
The best part about geometry is that the more you practice the more you'll remember. There are some standard results which you apply them right away. The more you practice, the more you can solve without using pen and paper.
Important topics in this section includes triangles and its properties, Congruence & Similarity of triangles, Chords & Tangents, Diagonals of a Quadrilateral, etc. However, in recent times, SSC is asking a lot of questions from Mensuration from 3 dimensional figures like pyramid, prisms etc
Trigonometry and Heights and Distance
This is the most productive topic that needs minimum time and can fetch you maximum marks. There are very few varieties of questions in Heights and Distances and you can easily remember all of them. Some of the formulas can be directly used and you can save your precious 5-10 minutes.
Moreover, if you are good in algebra, some concepts can be directly applied in solving Trigonometry questions.
It is very important to use trial and error in this section. Whenever there are questions that ask final vale of the trigonometric ratio, start by putting values of angles as 0 or 45 or 90. Moreover, eliminating the choices also helps sometimes.
Profit and Loss, Discount and Percentages
These are important topics in prelims as well as mains. Learn to solve percentages. You should know the fraction identity of percentages and many problems can be solved easily.
How to start preparing
SSC CGL is changing fast and the level is not same as it was 2 years ago. Also, being a computer based test now has its own perils. It is important to know the level of difficulty of questions. For example, DI in SSC CGL is far more easily compared with IBPS or SBI Exams however, Mensuration in SSC is a level higher than in IBPS or SBI exams. You must purchase previous year question books. The Advance Maths section of SSC CGL 2016 mains exams was much more calculative as compared to previous years. Even simple questions involved a lot of calculations. Learn tables and squares upto 30 and cubes upto 20. This will improve your calculations and save your time.
There are 100 high level questions that need to be solved in 120 minutes. To get into the top 100 rank, you should aim to solve the complete 100 questions in given time which can only be possible if you focus on two things simultaneously: Time and Accuracy.
Maths can only be improved by practicing. You need to start solving atleast 500 questions of a particular chapter only then you will know shortest way to get to the answer. Also when you are done with the syllabus, go for mocks. See how much you are able to solve in 120 minutes.
Practice Previous Year Questions
Sometimes, SSC repeats the questions. Try to solve previous 5 years questions of SSC CGL and keep marking the different concepts that you come across and keep revising it. Since it is online exam now, buy mocks from GradeUp where you can also compare your performance with 1000s of others.
On the Exam Day
After giving mocks, you will be able to know your strength and weakness. Learn the art of skipping the questions. Attempt the questions that you feel are easy and you are confident. In the first 60 minutes, you should be able to solve 55+ questions and should have seen around 70+ questions.
Last but not the least is to practice. Practice as much as you can. If you want to clear the exam, start doing and not just thinking. Getting good marks in an exam not only requires hard work, but smart work. You need to know what you should focus in a particular section and which the things that you should avoid.
The best way to prepare and assess your performance is by practicing on full length mock tests based on the updated pattern. To help you in your preparation, gradeup has launched a free SSC CGL Tier I mock test in Hindi & English which you can attempt through following link:
If you wish to be updated regarding the upcoming exams, their patterns, syllabus and other details, download Gradeup App for SSC CGL Preparation.. It provides free study material, updates of upcoming exams and mock tests for various competitive exams.
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Transcription
1 Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra. A student must be able to use symbolic algebra to represent and explain mathematical relationships in a cost analysis problem. A student must be able to judge the meaning, utility, and reasonableness of results of symbolic manipulations in the given real world context of the price of printing tickets. Common Core State Standards Math Content Standards High School Functions Interpreting Functions Understand the concept of a function and use function notation. F IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Interpret functions that arise in applications in terms of the context. F IF F IF Common Core State Standards Math Standards of Mathematical Practice MP.5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. MP. 7 Look for and make use of structure. Mat x 8 equals the well remembered 7 x x 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any 2012 Noyce Foundation
2 real numbers x and y. Assessment Results This task was developed by the Mathematics Assessment Resource Service and administered as part of a national, normed math assessment. For comparison purposes, teachers may be interested in the results of the national assessment, including the total points possible for the task, the number of core points, and the percent of students that scored at standard on the task. Related materials, including the scoring rubric, student work, and discussions of student understandings and misconceptions on the task, are included in the task packet. Grade Level Year Total Points Core Points % At Standard % 2012 Noyce Foundation
6 Printing Tickets Work the task and examine the rubric. What are the mathematical demands of the task? Look at student work on writing a formula. How many of your students put: (t/25) +1 (t+10)/25 (10t+1)/25 1t/25 11t/25 10t/25 25t/10 Other What big mathematical ideas are confusing for students? Which errors are related to order of operation or understanding symbolic notation? Which errors are related to not understanding the constant? Which errors are related to not understanding the rate of change? How often do your students work with making their own graphs? Do they make their own graphs from scratch or is most of the work done on graphing calculators? Do students have the habits of mind of making a table of values before completing a graph? Look at student work on graphs. Did their graphs match the values that would have fit their equations? Yes No How many understood that the graph should intercept the cost axis at $10 because of the constant or fixed cost? Look at student work on part three: How many of your students set the two equations equal to each other and solved for the unknowns? How many of your students used substitution of values most likely obtained from reading their graph? How many of your students picked the original values C=$2, t=25 or C=2t/25? Did your students think about more than 250 tickets and less than 250 tickets in part four? Did they think about large amounts and small amounts, but not quantify those values? Did their answers match their graphs? Did students think Sure Print would always be the best because you wouldn t have to pay the set up costs? What surprised you about student work? What are the implications for instruction? 69
7 Looking at Student Work on Printing Tickets Student A is able to write an equation to represent the costs of using Best Print and graph the equation. The student knew to set the two equations as equal to find the number of tickets when the costs would be the same. The student could quantify under what conditions it was better to use each print shop. Student A 70
8 Student B is also able to use algebra to solve the task. The student uses the idea of two equations with one unknown and solves the second equation for t. Then the student substitutes this solution for t into the first equation. Student B Student C is able to write an equation and graph it on page one of the task. Like many students, C seems to use the graph to find the point where the two values are the same and substitutes those values into the two original equations. Student C can t use the graph or information in part three to choose when to use each Print Shop. The student thinks that Susie should never choose Best Print because the $10 set up fee is too high. Student C 71
9 Student D is able to write an equation for Best Print and make its graph. The student seems to make notations at the bottom of the graph to help compare Sure Print with the new graph of Best Print. However the student isn t able to use this information to answer part three. The student does use the graph correctly to reason out part foour, determining the conditions for using each print shop. Student D 72
10 Student E is also able to write the equation and draw the graph. However the student uses only the costs of buying 25 tickets to compare each shop, rather than considering the values for buying a whole range of tickets and makes an incorrect conclusion. Student E Student F is unclear how to use symbolic notation to calculate the values for Best Print. However, the student seems to know the process and use it to make a correct graph. Notice all the points on the line, which seem to indicate calculated values. The student is able to look at the graph and see that costs are the same at 250 tickets. However when the student substitutes the values into the two equations, reasoning seems to shut down as the student finds values. The student takes the answers to the two equations after substitution and then divides. What might the student be thinking? What would you like to ask the student? 73
11 Student F 74
12 Student G writes a common incorrect equation. The graph does not match the results of calculations with that equation. Student G does not seem to understand the rate being used in this problem. The student sees the 25 in the two equations as the variable for number of tickets, not part of the rate 2 for 25 or 1 for 25. See the work in part three. The student seems to understand the general context of the problem and can reason (without quantity) when to use each print shop. Student G 75
13 Algebra Course One/Algebra Task 4 Printing Tickets Student Task Core Idea 3 Alg. Properties & Representations Compare price plans using graphs and formulae. Use inequalities in a practical context of buying tickets. Represent and analyze mathematical situations and structures using algebra. Write equivalent forms of equations, inequalities and systems of equations and solve them Use symbolic algebra to represent and explain mathematical relationships Judge the meaning, utility, and reasonableness of results of symbolic manipulations Based on teacher observation, this is what algebra students knew and were able to do: Write an equation for Best Print Draw a graph to match their equation Interpreting graphs of two equations to determine best buy under different conditions Areas of difficulty for algebra students: Understand how to use symbolic notation to represent a context Find a table of values before drawing a graph Using algebra to solve for 2 equations with 2 unknowns 76
14 The maximum score available on this task is 9 points. The minimum score for a level 3 response, meeting standards, is 4 points. More than half the students, 60%, could write an equation to represent the cost of buying tickets at Best Print. Almost half the students, 40%, could also graph the cost of Best Print. Some students, could find when it was cheaper to use Best Buy or Sure Print. 8.5% of the students could meet all the demands of the task including using algebra to find the point where the costs for Best Buy and Sure Print are the same. Almost 40% of the students scored no points on this task. 90% of the students with this score attempted the task. 77
15 Because of the number of students scoring zero, their thinking is documented below: Common Equation Errors: (t+10)/25 (10t +1)/25 1t/25 11t/25 10t/25 25t/10 6% 3% 14% 16% 11% 3% Graphing Errors: 58% of the graphs did not match the equation the student had written for part 1. 21% of the graphs had lines parallel to Sure Print 43% of the graphs for Best Print went through the origin (0,0) which would only be true if there were no constant ($10 set up) Finding values for C= and t+ : No answer 41% Picking 25 tickets with some other value: 30% Picking the best print shop: Sure Print and matches their graph: 24% Sure Print, doesn t match their graph: 6% Sure Print, some other reason: 11% Best Print, matches their graph: 6% Best Print, doesn t match their graph: 8% Best Print, other reasons: 8% Printing Tickets for all students Points Understandings Misunderstandings 0 90% of the students with this 40% of all students scored no point on this score attempted the task. 1 Students could write an equation to represent the cost of tickets using Best Print. 3 Students could write and graph an equation for Best Print. 4 No clear pattern. 7 Students could write and graph the equation. They could determine when the costs were the same for both companies. They could explain which company to use in different situations. 9 Students could meet all the demands of the task including using algebra to find when the costs were equal. task. See analysis of their work above. Students did not understand how to use the constant of $10. Some treated it as a variable, 11t over 25 or 10t/25 or (10t +1)/ 25. Some students struggled with order of operations (t+10)/25. Many students did not make graphs that matched values that could be obtained from their equations. There was no evidence of making a table of values before making their graphs. Students could not use algebra to solve for when the costs were the same. Most students used the information from their graphs and substituted the values into the two equations. 78
16 Implications for Instruction Students at this level need more opportunities to use algebra in a practical situation. Students should have practice making a table of values to help them graph equations. They should also understand how a constant effects the graph and be able to use the formula to think about slope. Some students at this level are still struggling with understanding the meaning of variables. They see the letters or symbols as standing for labels. Others think that an equation is only for finding one specific value. They don t understand that the letter represents a quantity that can vary or change. Students need more experience with solving problems in context that promote discussion about how the variable may change and why. They need to connect the equation to a wide range of possibilities, to a representation of a more global picture of a situation. These nuances do not come through practice with just symbolic manipulation. A few students struggle with the basic algebraic notation around order operations, combining algebraic fractions and whole numbers, and solving equations with divisors or fractional parts. There are many situations where it is important to find the breakeven point or place where two functions intersect. Students should be familiar with these types of situations and be comfortable setting the two equations to equal each other. These problems might include choosing the best rate for a gym membership, picking a cell phone plan, or ways to price a store item with different costs having different volumes of sales. (See MAC tasks th Grade: Gym, th :Picking Apples) 79
Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations
Performance Assessment Task Sorting Functions Grade 10 This task challenges a student to use knowledge of equations to match tables, verbal descriptions, and tables to equations. A student must be able
Performance Assessment Task Carol s Numbers Grade 2 The task challenges a student to demonstrate understanding of concepts involved in place value. A student must understand the relative magnitude of whole
Performance Assessment Task Symmetrical Patterns Grade 4 The task challenges a student to demonstrate understanding of the concept of symmetry. A student must be able to name a variety of two-dimensional
Performance Assessment Task Swimming Pool Grade 9 The task challenges a student to demonstrate understanding of the concept of quantities. A student must understand the attributes of trapezoids, how to
Performance Assessment Task Squares and Circles Grade 8 The task challenges a student to demonstrate understanding of the concepts of linear equations. A student must understand relations and functions,
Performance Assessment Task Which Is Bigger? Grade 7 This task challenges a student to use knowledge of measurement to find the size of objects in a scale drawing. A student must be able to solve problems
Performance Assessment Task Conference Tables Grade 9 This task challenges a student to use knowledge of symbolic notation and functions to write an equation for two growing patterns. A student must be
Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the
Performance Assessment Task Aaron s Designs Grade 8 This task challenges a student to use transformations, reflections and rotations on a coordinate grid. A student must be able to quantify a transformation
Performance Assessment Task Gym Grade 6 This task challenges a student to use rules to calculate and compare the costs of memberships. Students must be able to work with the idea of break-even point to
Performance Assessment Task Graphs (2004) Grade 9 This task challenges a student to use knowledge of functions to match verbal descriptions of a context with equations and graphs. A student must be able
Performance Assessment Task Mixing Paints Grade 7 This task challenges a student to use ratios and percents to solve a practical problem. A student must use knowledge of fractions and ratios to solve problems
Performance Assessment Task Number Trains Grade 4 The task challenges a student to demonstrate understanding of the concepts of factors and multiples. A student must develop fluency with basic multiplicative
Performance Assessment Task Coffee Grade 10 This task challenges a student to represent a context by constructing two equations from a table. A student must be able to solve two equations with two unknowns
Performance Assessment Task Picking Fractions Grade 4 The task challenges a student to demonstrate understanding of the concept of equivalent fractions. A student must understand how the number and size
Algebra 2, Quarter 1, Unit 1.1 Quadratic Functions and Their Graphs Overview Number of instructional days: 6 (1 day = 45 minutes) Content to be learned Create a graph of a quadratic function from a table
Performance Assessment Task Pizza Crusts Grade 7 This task challenges a student to calculate area and perimeters of squares and rectangles and find circumference and area of a circle. Students must find
Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to
Performance Assessment Task Hexagons Grade 7 task aligns in part to CCSSM HS Algebra The task challenges a student to demonstrate understanding of the concepts of relations and functions. A student must
Performance Assessment Task Bikes and Trikes Grade 4 The task challenges a student to demonstrate understanding of concepts involved in multiplication. A student must make sense of equal sized groups of
Performance Assessment Task Hopewell Geometry Grade 10 This task challenges a student to use understanding of similar triangles to identifying similar triangles on a grid and from dimensions. A student
REPRODUCIBLE Figure 4.4: Evaluation Tool for Assessment Instrument Quality Assessment indicators Description of Level 1 of the Indicator Are Not Present Limited of This Indicator Are Present Substantially
Grade 3 Mathematics, Quarter 1, Unit 1.1 Number System Addition and Subtraction, Place Value and Patterns in Addition Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned
Parent Functions and Transformations Overview Number of instruction days: 5 7 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Recognize the properties and characteristics
Grade 5 Mathematics, Quarter 4, Unit 4.1 Line Plots Overview Number of Instructional Days: 10 (1 day = 45 60 minutes) Content to be Learned Make a line plot to display a data set of measurements in fractions
Performance Assessment Task Scatter Diagram Grade 9 task aligns in part to CCSSM grade 8 This task challenges a student to select and use appropriate statistical methods to analyze data. A student must
Performance Assessment Task Rhombuses Grade 10 This task challenges a student to use their knowledge of Pythagorean theorem to show that given rhombuses are similar. A student must use knowledge of proportional
Performance Assessment Task Hexagons in a Row Grade 5 This task challenges a student to use knowledge of number patterns and operations to identify and extend a pattern. A student must be able to describe
Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for
Performance Assessment Task Houses in a Row Grade 3 This task challenges a student to use knowledge geometric and numerical patterns to identify and continue a pattern. A student must be able to use knowledge
Grade 6 Mathematics, Quarter 4, Unit 4.1 Polygons and Area Overview Number of instructional days: 12 (1 day = 45 60 minutes) Content to be learned Calculate the area of polygons by composing into rectangles
A Correlation of Common Core Course 3, 2013 to the Standards for Mathematical Practices Standards for Mathematical Content Grade 8 to the Table of Contents Mathematical Practices...1 The Number System...8
Performance Assessment Task Incredible Equations Grade 2 The task challenges a student to demonstrate understanding of concepts involved in addition and subtraction. A student must be able to understand
The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems
Grade 5 Mathematics, Quarter 4, Unit 4.1 Representing and Interpreting Data on a Line Plot Overview Number of instructional days: 8 (1 day = 45 60 minutes) Content to be learned Make a line plot to display
Performance Assessment Task Baseball Players Grade 6 The task challenges a student to demonstrate understanding of the measures of center the mean, median and range. A student must be able to use the measures
Grade level/course: 2 nd Grade/Math Trimester: 2 Unit of study number: 2.6 Unit of study title: Addition and Subtraction Using Strategies Number of days for this unit: 10 days (60 minutes per day) Common
Geometry, Quarter 1, Unit 1.1 Line Segments, Distance, and Midpoint Overview Number of instructional days: 8 (1 day = 45 60 minutes) Content to be learned Know the precise definition of line segment, based
Grade 1 Mathematics, Quarter 1, Unit 1.1 Telling Time to the Hour Overview Number of instructional days: 3 (1 day = 45 60 minutes) Content to be learned Correctly tell time in hours using analog and digital
Common Core State Standards for MATHEMATICS Work Session #1a Handout #1a Mathematics Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics
Performance Assessment Task High Horse Grade 2 The task challenges a student to demonstrate understanding of measurement. Students must measure an object using two different units and compare their results.
Being Building Blocks This problem gives you the chance to: work with area and volume Barbara s baby brother, Billy, has a set of building blocks. Each block is 2 inches long, 2 inches wide, and 2 inchesMA.PS.1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.
Mathematical Practices are listed throughout the grade level document in the 2nd column to reflect the need to connect the mathematical practices to mathematical content in instruction. MP.1. Make sense
CORE High School Math Academy Alignment with Core State Standards CORE High School Math Academy emphasizes throughout each day the importance of teaching along the five proficiency strands identified by
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
Performance Assessment Task Peanuts and Ducks Grade 2 The task challenges a student to demonstrate understanding of concepts involved in addition and subtraction. A student must be fluent with addition
COMMON CORE STATE STANDARDS FOR MATHEMATICS The Standards for Mathematical Practice is a document in the CCSS that describes different types of expertise students should possess and mathematics educators
Performance Assessment Task Garden Design Grade 3 This task challenges a student to use understanding of area and count squares to find the area of shapes on a grid. A student must be able to compare the
Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed
Performance Assessment Task Marble Game task aligns in part to CCSSM HS Statistics & Probability Task Description The task challenges a student to demonstrate an understanding of theoretical and empirical
The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems | 677.169 | 1 |
Math 098 - Intermediate Algebra
Text:
Intermediate Algebra with Early Functions. Second edition. James W. Hall. Copyright 1995,
International Thomson Publishing. (Required)
Student's Solution Manual. (Optional)
Student Audience:
Most students going on to advanced courses in Mathematics and those wishing to study technical
programs will take this course. This is the RCC entrance course for Math.
Prerequisite:
The prerequisite is successful completion of both Math 091 (Basic Algebra) and Math 095
(Geometry) or the successful completion of a placement test.
Course Description:
Mathematics 098, Intermediate Algebra, includes instruction in algebraic topics common to the
standard college Intermediate Algebra course. General objectives in the course are to: identify,
develop, and solve problems related to real world situations; identify and use various problem
solving strategies; interpret tabular and graphical data to solve physical problems; manipulate
mathematical sentences numerically, symbolically, and graphically; compute with radicals,
exponents, and complex numbers; use technology appropriately in problem solving and in
exploring and developing mathematical concepts; use the process of mathematical discovery
(conjecture, testing, refinement, more testing, and final statement of results).
Course Objectives:
Upon successful completion, the student will demonstrate proficiency and understanding in the
following topics: Review of real number operations and properties; First degree equations and
inequalities; absolute value equations and inequalities; Elementary operations with polynomials
and factoring; Operations with algebraic fractions and solving fractional equations; Integer and
Rational exponents; Simplification of radicals; Operations with complex numbers; Addition,
subtraction, multiplication, division, and whole number powers of I; Second degree equations and
inequalities; Graphing lines, other graphs, distance formula and circles; Functions - definition,
linear functions, other functions; Systems of linear equations and inequalities (using elimination
and substitution). exam grade will be zero.
Grading Policy:
There will be several one hour examinations and a comprehensive final examination. Announced
and unannounced quizzes may be given. Various homework exercises may be used in grading.
Note: Homework is essential to the study of mathematics. Letter grades will be assigned to final
adjusted scores as follows: A=90-00%; B=80-89%; C=70-79%; D=60-69%; F=0-59%.
Consideration will be given to such qualities as attendance, class participation, attentiveness,
attitude in class, and cooperation to produce the maximum learning situation for everyone.
Any student who stops attending without dropping will receive a grade of F.
A notebook should be kept which contains every problem worked in class as well as any
comments that are appropriate. In general, it should contain everything written on the
chalkboard. Be sure to bring your notebook if you come to the instructor or a tutor for help.
Type of Instruction:
Lecture, discussion, problem solving, and group work will be used. Students should come to
class with a prepared list of questions.
Calculators:
Calculators may be used to do homework. Calculators may be used on exams and/or quizzes in
class unless otherwise announced. The calculator should be a scientific calculator capable of
doing logarithms. A graphing calculator, such as the TI-82, is also There will be a stapler and paper
punch available in the classroom.
Additional Help:
Office hours will be announced. The student is encouraged to additional help when the material
is not comprehended. Mathematics is a cumulative subject; therefore, getting behind is a very
difficult situation for the student.
If your class(es) leave you puzzled, the Study Assistance Center is a service that Richland
Community College offers you. It is available free of charge to all RCC students.
There are video tapes on reserve in the Learning Resources Center to accompany this course.
These are suggested if you miss a lesson, or want additional explanation. | 677.169 | 1 |
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Description
Algebra is fundamental to the working of modern society, yet its origins are as old as the beginnings of civilization. Algebraic equations describe the laws of science, the principles of engineering, and the rules of business. The power of algebra lies in its efficient symbolic representation of complex ideas, and this also presents the main difficulty in learning it. It is easy to forget the underlying structure of algebra and rely instead on a surface knowledge of algebraic manipulations.
• A Balanced Approach: Form, Function, and Fluency- students given an ability to recognize algebraic form and an understanding of the purpose of different forms.
• Restoring Meaning to Expressions and Equations- after introducing each type of function-linear, power, quadratic, exponential, polynomial- the authors study the basic forms of expressions.
• Maintaining Manipulative Skills: Tool Section- placed at the end of chapters to which they are particularly pertinent
In response to reviewer comments and suggestions, we have made the following changes since the preliminary edition.
• Added four new chapters on summation notation, sequences and series, matrices, and probability and statistics.
• Split the initial chapter reviewing basic skills into three shorter chapters on rules and the reasons for them, placed throughout the book.
• Added Focus on Practice sections at the end of the chapters on linear, power, and quadratic functions. These sections provide practice solving linear, power, and quadratic equations.
• Added material on radical expressions to Chapter 6, the chapter on the exponent rules.
• Added material on solving inequalities, and absolute value equations and inequalities, to Chapter 3, the chapter on rules for equations.
• Added more exercises and problems throughout
A Balanced Approach: Form, Function, and Fluency Form follows function. The form of a wing follows from the function of flight. Similarly, the form of an algebraic expression or equation reflects its function. To use algebra in later courses, students need not only manipulative skill, but fluency in the language of algebra, including an ability to recognize algebraic form and an understanding of the purpose of different forms.
Restoring Meaning to Expressions and Equations After introducing each type of function—linear, power, quadratic, exponential, polynomial—the text encourages students to pause and examine the basic forms of expressions for that function, see how they are constructed, and consider the different properties of the function that the different forms reveal. Students also study the types of equations that arise from each function.
Maintaining Manipulative Skills: Review and Practice Acquiring the skills to perform basic algebraic manipulations is as important as recognizing algebraic forms. Algebra: Form and Function provides sections reviewing the rules of algebra, and the reasons for them, throughout the book, numerous exercises to reinforce skills in each chapter, and a section of drill problems on solving equations at the end of the chapters on linear, power, and quadratic functions.
Students with Varying Backgrounds Algebra: Form and Function is thought-provoking for well-prepared students while still accessible to students with weaker backgrounds, making it understandable to students of all ability levels. By emphasizing the basic ideas of algebra, the book provides a conceptual basis to help students master the material. After completing this course, students will be well-prepared for Precalculus, Calculus, and other subsequent courses in mathematics and other disciplines. | 677.169 | 1 |
Suswantari_Mathematics_07305141036
Kamis, 15 Januari 2009
It's difficult to communicate mathematics language to the other. That is not as easy as read or understand any topic in mathematics. We have to be smart to find the right words, not so hard word, so that others can understand what we said. The difficulties to communicate this mathematics language become the challenge for university students of Mathematics in Yogyakarta State University to complete an English assignment lectured by Dr. Marsigit, M. A.. We have to communicate any topic about mathematics, absolutely in English, in pairs. I am with Ardhita Kirana Rukmi. I take The Polynomials from mathematics book of 11th grade Senior High School, second semester, published by Erlangga. Firstly, I give her the definitions of the polynomials. Then, I show her an example of the polynomials write. She seems not in trouble in understanding what I said. So, I move to give her several questions about the definitions of the polynomials as follows: Mention the variable, the degree, and the coefficients for each following polynomials: a. (4+3t-2t^2+t^3+10t^4-2t^5) b. (2x^3+5x^2-10x+7) c. (b^8-6b^7+5b^6-16) She can do it well. I told her the next subsection about the value of the polynomials after the examples. There are two methods to find the value of the polynomial in this subsection. First method is the substitution and the second is the scheme method. From this two methods, Ardhita like the first method better. She said that she always used the substitution method since the first time she got it to find the value of the polynomials. Here's the example: Determine the value of the polynomial f(x)=x^3+3x^2-x+5 if x is replaced by x=m (m element of R). Answer: For x=m we obtain f(m)=m^3+3m^2-m+5. I get a little bit difficulties for the scheme method because the explanation from the book uses an alphabet so it seems so abstract. I told Ardhita step by step on scheme method. Unfortunately, she's not so understands about what I give. It maybe caused by my language that difficult to understand. But, she said that she had understands after I show the picture of the scheme. I only take the subject matter until the second subsection. I give the exercise after finishing explaining her to see how far she knows about the polynomials. I give her three questions: 1. With the substitution method, find f(1,y) if f(x,y)=x^3y^4+xy^3+y+2x^2+3 2. Find f(2,-1) if f(x,y)=x^2y+y^2-2x+10 with the substitution method 3. Find the value of the polynomials from number 1 and two with the scheme method She did the two first questions well. For the last question using the scheme method she needs a little longer time than the first. From this activity I can conclude that Ardhita understanding about the polynomials better than before. The first reason is that she had learned the lesson about the polynomials since junior high school. The second reason is the definition of the polynomial reminding her to the very beginning we learned it. Or on the words it will remind all of the things that maybe passed by her. Scheme method rarely used by Ardhita to solve the problems. She keen on the substitution method better because this method always she use. It becomes one problem to understand the scheme method quickly. She said that she uses the scheme method in factoring polynomials, not to find the value of the polynomials. The main challenge for me is in communicate this polynomials topic in English. A little number of mathematics vocabularies is the problem that makes our communication heard not so smooth. Mathematics term maybe rarely heard. But it must be an ordinary thing for mathematics student like me. Other problem that I face is translating this matter. I take it from the book in Indonesian. There are several term that not existing in the dictionary. I ask to my friend for sometimes or browse from the internet to handle it. The problems can be cleared then. This activity is so good and gives so many advantages for student to know how far our communication capability in English about mathematics. I motivated to be more diligent to learn and learn, English especially, because nowadays we have to being a part of an international society from the globalization.
1. Pre Calculus Graph Let's begin by discussing the graph of a rational function which can have discontinuities. A rational function has a polynomial in the denominator which means you are dividing by something that is valuable quantity. It's possible that some value of x will meet to division be zero. Example: If When It also mean that the graph of this Review A When For Removable singularity appears simplest missing points on the graph when x leads to 0/0. For this kind of a break, if you factor and simplify the rational function, the division by zero can be avoided.
There are two conditions : 1. X goes to positive or negative infinity 2. Limit involves a polynomial divide by a polynomial For example : Lim x3 + 4 = ~ x2 + x+1 X→ ~ (Because highest power of x in number is 3 greater than highest power in denominator) since all the number are positive and x is positive infinity The key to defermining limits by inspection is in looking at powers of x in the numerator ana the denominator. To apply these rules : - must be divining by polynomial - x has to be approaching infinity
1. First shortcut rule If the highest power of x in numerator is 3 greather than highest power in denominator) since all the number are positive and x is going to positive infinity if you can't tell it the answer is positive or negative: - substitute a large number for x - see if you end up with a positive or negative number - whatever sign you get is the sign of infinity for the limit.
2. Second shortcut rule: If the highest power of x is in the denominator, then limit is zero Lim x2 + 3 x → ~ x3 + I , x2 + 3 = highest power of x in numerator is 2 x3 + 1 = highest power of x in denominator is 3
3. Last shortcut rule Used when : highest power of x in numerator is same as highest power of x in denominator. Lim = the quotient on the efficient on the two highest powers x → ~ -Remember : coefisient → the number that goes with a variable Ex : 2 is the coefficient of 2x2 75 is the coefficient of 75 x 4 Sho that is no way if x = 3
When you see the result and also tell you direction be possible factor top and bottom of rasional function and simplify
Algebraic form had been learned at junior High School which is discussing about the definition of a term, factor, coefficient, and constant. Monomial, binomial, and polynomial in same or different variable, solving the operations of the polynomial, solving the division of the same term, and factoring algebraic terms had been studied too. That matter will be discussed again deeper, then will be developed to the division, the polynomials, remaining theorem, and the roots of the polynomials. The basic competent of the "Polynomial" is using the division algorithm, remaining theorem, and factoring theorem on problem solving Realization of the competent will be show through study result : using polynomials division algorithm to decide the division result and division remain, using the theorem and the factor to solve the problems, and proof the remaining and the factoring theorem. To support the success of the competent in this chapter, the indicators of the success are, the student can: - explain the polynomials division algorithm - decide the degree of division result polynomials remain by linear or quadratic form - determine the division result and division remain by linear or quadratic form - determine the polynomials division remain by linear or quadratic form using the remaining theorem - determine the linear factor of the polynomials using factoring theorem - proof the remaining theorem and the factoring theorem
5-1 Definitions of polynomials, the value of the polynomials, and polynomials operations
5-1-1 Definition Look at there algebraic form: (i) x2-3x+4 (ii) 4x3 + x2-16x+2 (iii) X4+3x3-12x2+10x+5 (iv) 2x5-10x4+2x3+3x2+15x-6 Those algebraic forms are called polynomials in x. The degree/ exponent of the polynomial in x decided by the highest exponent of x. For example: (i) x2-3x+4 is a second degree polynomial because the highest exponent of x is 2 (ii) 4x3 + x2-16x+2 is a third degree polynomial because the highest exponent of x is 3 (iii) X4+3x3-12x2+10x+5 is a fourth degree polynomial because the highest exponent of x is 4 (iv) 2x5-10x4+2x3+3x2+15x-6 is a fifth degree polynomial because the highest exponent of x is 5 So, the polynomial's degree in x, generally wrote as:
anxn + an-1xn-1 + an-2xn-2+…+a2x2 + a1x + a0
Where: - an, an-1, an-2, …, a2, a1, a0, are real number with an 0. an is the coefficient of xn, an-1 is the confident of xn-1, an-2 is the coefficient of xn-2, …, and soon, a0 called as a constant. - n is a counting number which represent the degree of the polynomials the terms of the polynomials above are beginning by the term that the variable has he highest power ansn, then followed by the terms with decreasing power of x, an-1xn-1, an-2xn-2, …, a2x2, a1x1, and ended by the constant 90. The polynomials written on that way called arranged in decreasing power rule on variable of x. Remember that the variable must not be x, but can be a,b,c…, y and z. For illustration, look at these polynomials: a) 6x2-3x2+10x+4 is a third degree polynomial in x. The coefficient of x3 is 6, for x2 15-3, for x is 10, and the constant is 4. b) 9y4-y3+5y2-2y-13 is a fourth degree polynomial in x. The coefficient of y4 is 9, for y3 is -1, for x is 5, for y is-2, and the constant is -13. c) (x-1)(x+1)2=x3+x2-x-1 is a third degree polynomial in x. The coefficient of x3 is 1, for x is -1, and the constant is -1. d) (t+1)2(t-2)(t+3)=t4+3t3-3t2-11t-6 is a fourth degree polynomial in t. The coefficient of t4 is 1, for t3 is 3, for t3 is -3, for t is-11, and the constant is -6. Polynomials above are polynomials in one variable called a univariable polynomials. The polynomials in more than one variable called a multivariable polynomials. For illustration, look at following polynomials. a) x3+x2y4-4x+3y2-10 is a polynomial in two variable x and y. this is a third degree polynomial in x or fourth degree polynomial in y b) a3+b3+c3+3ab+3ac-bc+8 is a polynomial in three variables a, b, and c. this is a third degree polynomial in a, b, or c.
5-1-2 The value of the polynomials The polynomials can be written in the form of function of the variable, based on the fact that polynomial is an algebraic form in a variable. Polynomials in x can be written as a function of x. for example, general form of the polynomials can be state in a function form as:
f(x)=anxn+ an-1xn-1+ an-2xn-2+…+a2x2+a1x+a0
Notes: Polynomial function above is state in f (x), sometimes state as: - S(x) that shows polynomial function in x, or - P(x) that shows polynomial function in x
Denote the polynomials as a function in x, then the polynomials value can be determined. Generally, polynomials value of f(x) for x = k is f(k) where k is a real number. Then, the value of f(k) can be found with two methods:
Based on the description above, the value of the polynomials for a certain variable can be found by the rule of substation method as follows: The polynomials value of f(x)=anxn+ an-1xn-1+ an-2xn-2+…+a2x2+a1x+a0 for x = k (k real numbers) determined by: f(x)=an(k)n+ an-1(k)n-1+ an-2(k)n-2+…+a2 (k)2+a1 (k)+a0
The value of f(x) for x = k obtained by substitute the value of k to the variable of x on the polynomials of f(x). Hence, counting the polynomials value as above called as a substitution method. Look at these example to understanding the way to counting polynomials value with substitution method.
Based on the last equation, we can see that the value of f(k) can be found step by step with an algorithms as follow: - First step: Multiply a4 with k, then sum the result with a3 a4k + a3 - Second step Multiply the result of the first step (a4k + a3) with k, then sum the result with a2 (a4k + a3)k + a2 = a4k2 + a3k + a2 - Third step Multiply the result of the first step (a4k+a3k+a2 ) with k, then sum the result with a1 (a4k+a3k+a2 )k+a1= a4k2 + a3k + a2k+a1 - Fourth step Multiply the result of the first step (a4k+a3k+a2k+a1) with k, then sum the result with a0 (a4k+a3k+a2k+a1)k+a0= a4k+a3k+a2k+a1k+a0 The result of this fourth step is the value of f(x)= a4k+a3k+a2k+a1k+a0 for x=k The processes of the algorithms above can be displayed on a scheme x = k is wrote in the first row of the scheme, then followed by polynomials coefficients. These coefficients arranged from the highest exponent coefficients to the smallest exponent. Look at this scheme:
Finding the value of the polynomials as above called as a scheme method. This name is given because of the scheme that used.
Notes: (1) Substitution method is suitable to find the value of the polynomials with a simple form for a small value and integer of x (2) Scheme method can be used o find the value of all forms of the polynomials and for arbitrary x E
Here are the applications examples to find the value of polynomials with scheme method: Example 3: Find the value of these polynomials with scheme method a) f(x) = x4-3x3+4x2-x+10 for x = 5 b) f(x) = x5-x2+4x-10 for x = 2 Answer: a) The coefficients of f(x) = x4-3x3+4x2-x+10 are a5=1, a4=a3=0,a2=-1,a1=4, and a0 = -10 For x =5, so k = 5.The scheme is shown on picture 5-2a. Based on the scheme, the value of f(x) = x4-3x3+4x2-x+10 for x=5 is f(5) = 355 b) The coefficients of f(x) = x5-x2+4x-10 are a5=1, a4=a3=0,a2= -1,a1=4, and a0 = -10 For x =2, so k = 5.The scheme is shown on picture 5-2a. Based on the scheme, the value of f(x) = x5-x2+4x-10 for x=2 is f(2) = 26
5 a4 a3 a2 a1 a0 + + + + 5 10 70 345 1 2 14 69 355
2 a5 a4 a2 a1 a0 + + + + 2 4 8 36 1 2 4 ¬18 26
Example 4: Find the value of this polynomial with scheme method: f(x,y)=x2y+x3 y2+x2+3y+2 for x = 2 Answer: To find the value of f(x,y) for x=2, f(x,y) is considered as a polynomial in x with the form of decreasing exponents: f(x,y)=y2x3 + (y+1)x2+(3y+2) The coefficients of f(x,y) are a3=y2, a2 = y+1, a1=0, and a0 =3y+2. x=2 means k=2. Based on the scheme, the value of f(x,y)=x2y+x3 y2+x2+3y+2 for x = 2 is f(2,y)=8y+7y+6 The value of f(x) = x5-x2+4x-10 for x=2 is f(2) = 26
The boy are try to motivate us to build our confidence. Anyone can do anything, can be anything, and become anything. No matter where we come from, we are just the same each other. We got to trust to ourselves that we can do what we want to do. When there is no belief in our heart, we will be nothing. The key to be confident, to be survive in our life is be believe in ourselves. We can be success in our life if we do trust that we can do anything and be anything. No one have no belief. The boy is a good motivator. He can persuade everybody around him to be come in his speech.
Revision:
DO YOU BELIEVE?
There is a boy named Dalton Sherman. He came from Charles Rice Learning Center. He tried to promote the Charles Rice Learning Center to the audience in Dallas ISD. Firstly, he asked to the audience, do they believe that he can stand up on the stage, fearless, and talk to them. He said that everyone must believe in him because there is some deals that he can do anything, be anything, and become anything. Then, he asked to the Dallas ISD, do they believe in his classmate and do they believe that everyone of them can graduate for workplace or college. He said, they better do believe, because next week they will show up the Dallas ISD that they can reach their highest potential, no matter where they come from, they better not give up on them. The audiences are the one who feed them and who wipe their tears. They are the one who love them when no one does. Next, he asked do they believe in their colleagues. They came to the audience's school because they want to be developed also. No matter what they are, he needs them. He tried to persuade then to trust to their colleagues, so they will do to. What are they doing is not only for his generation, but to the next generation also. The boy and his classmate need the audience more than ever. So, the audience needs to believe in themselves. Then he asks do they believe that every child in Dallas needs to be ready for college? They must believe that they can do. They have to believe in their colleagues, in themselves, and in their goals. Finally, he say thank you to them who does so many things for him and others. They have to believe in him because he believes in himself. The audience helped him get to where he is today.
Minggu, 21 Desember 2008
The figure shows the graph of y=g(x), in the function h is defined by h(x)=g(2x)+2, what is the value of h(1)?
Answer: We are looking for h(1). The first information is the graph. The next piece of the information is that h(x)=g(2x)+2. What is h equal when x is equal to 1? So let's substitute 1 into h(x)=g(2x)+2 àh(1)=g(2)+2. Now, how we figure out g(2) or what is g when x is equal to 2? We have the graph above. When x is equal to 2, y is equal to 1. Then g is equal to 1 when x is equal to 2. So, h(1)=g(2)+2 become h(1)=1+2 and we got h(1)=3.
The next problem is another question 13 on page 534:
Let the function f be defined by f(x)=x+1, if 2f(p)=20, what is the value of f(3p)?
Answer: We are looking for f(3p) or in another word what is f when x is equal to 3p? The first piece of information is equation of f(x)=x+1 and the next information is 2f(p)=20. To figure out what is f when x=3p, we need to figure out what p is. Let's start with the equation of2f(p)=20. Divide both side by 2, we get f(p)=10. Then, f(p) is just what is function f(x) when x is equal to p. Write that equation. Plug in p in the equation f(x)=x+1àf(p)=p+1=10àp=9. That is not the answer because we are looking for f when x=3p. So if we take p=9 to x=3p, we get x=27. We have the equation for function f which is x+1. We know that x=27, so f(27)=27+1=28.
Let' move to the next question. Question 17 on page 412:
In the xy-coordinate plane, the graph of x=y^2-4 intersect line l at (0,p) and (5,t). What is the greatest possible value of the slope of l?
Answer: We are looking for the greatest slope (m). In thexy-coordinate plane, the graph of x=y^2-4 is:
x=y^2-4 intersect line l at two points (0,p) and (5,t). It does not define exactly where those points are. We have the general idea of the line l on this graph.
Rewrite those two points and put it in a little table to help us.
What is the greatest slope? How do we know about the slope. We know that the slope at the line is going to be m=(y2-y1)/(x2-x1). We can put the values of x and y from the coordinate table that apply to line l. So the slope is going to be m=(t-p)/(5-0)=(t-p)/5. If we want to maximize the slope, we need to maximize the numerator (t-p). How we figure out (t-p)? We have the equation x=y^2-4 and the coordinates (0,p) and (5,t) applied to the equation. The points are the intersection of the graph and the line. So we can plug in the coordinates into the equation of x=y^2-4.
2.Factoring Polynomials
One way to define factor of the polynomials is the rule of algebraic long division that looks like the long division you learn, only harder. For an example, let's try to see:
Is x-3 a factor of x^3-7x-6?
Answer: When dividing x-3 into x^3-7x-6, first setup the problem like a long division problem from elementary school. That is, you dividing x-3 into x^3-7x-6. Now the zero is in there because there is no second degree term. Now you must ask yourself, what times x gives you x^3? Of course it is x^2. So you write x^2 as the part of the quotient and then multiply x-3 by x^2 which gives you x^3-7x, which you subtract from x^3-0x^2 to get 3x^2. Bringing down the next term -7x, you have 3x^2-7x.
Now we begin again dividing x-3 into 3x^2-7x. Just looking at the first term x goes into 3x^2, 3x. The next part of the answer is 3x. Multiply x-3 by 3x for a product of 3x^2-9x. Subtracting with 3x^2-7x you are left with 2x-6. Now we see that x-3 divides evenly into 2x-6 which equal 2, with no remainder. So the solution to the long division problem x^3-7x-6 divided by x-3 is x^2+3x+2.
Since x-3 divided into x^3-7x-6 evenly with no remainder then x-3 is a factor of x^3-7x-6. The quotient which is x^2+3x+2 is also a factor of x^3-7x-6. We know that x^3-7x-6=(x-3)(x^2+3x+2). The quadratic expression (x^2+3x+2) can be factored into (x+1)(x+2). So x^3-7x-6=(x-3)(x+1)(x+2). Setting the factored form of the equation x^3-7x-6 to zero, we get 0=(x-3)(x+1)(x+2). Those either x-3=0, or x+1=0, or x+2=0. Solving all of this equation for x, we get x=3, x= -1, and x= -2. The roots to x^3-7x-6 are 3, -1, and -2.
Now there are three roots for the third degree equation of x^3-7x-6. On the quadratic or second degree equation we look that always have at most two roots. A fourth degree equation would have four or fewer roots and so on. The degree of a polynomial equation always limits the number of roots.
Let's summarize the long division process for third order polynomial:
1.Find a partial quotient of x^2, by dividing x into the first term x^3 to get x^2.
2.Multiply x^2 by the divisor and subtract the product from the dividend.
3.Repeat the process until you either "clear it out" or reach a remainder.
3.Pre-Calculus Graph
Let's begin by discussing the graph of a rational function which can have discontinuities. A rational function has a polynomial in the denominator which means you are dividing by something that is valuable quantity.
It's possible that some value of x will meet to division be zero. Example:
If
When
It also mean that the graph of this
Review
A
When
For
Removable singularity appears simplest missing points on the graph when x leads to 0/0. For this kind of a break, if you factor and simplify the rational function, the division by zero can be avoided.
4. Inverse Function
We will talk about the inverse function. To talk about it, we need to review the definition of the function. If we have the relation of f(x,y)=o, then function y=f(x). That is 1.1 function if we can represent it in x=g(y). The function y=f(x) is "VLT" (Vertical Line Test). Every vertical line intersect the graph in at most one point. The 1.1 function satisfy "HLT" (Horizontal Line Test).
If we look at function y=x^2, this is not 1.1 function because if we graph any horizontal line on the graph, we get two intersection points.
To make it as a 1.1 function, we need a domain of 0<=x, so the graph will be:
We can figure out the other squared root of the left side so if we have function of 1.1 then the function is "invertible".
Let's start out with the function y=2x-1. And let's look at the graph of that function.
This is a straight line with y intersect (-1) and x intersect (1/2). Look at the line y=x. That line intersect the graph of y=2x-1. So, we get x=2x-1à 1+x=2xà 1=x. So, the intersection of the line y=x and y=2x-1 with x=1 is (1,1).
See this relation for y=x. Write 2x-1=yà 2x=y+1 àx=(y+1)/2 àx=(y/2)+1/2. Changing x to y in the last equation, we get y=(x/2)+1/2. Looking back on the graph, then we get another line. That line contain the point (1,1). The x-intercept is (0,-1) and the y-intercept is (0,1/2). So, the invers line to the given line y=2x-1 passes through the same point.
We have f(x)=2x-1 and on the other hand, we have g(x)=(x/2)+1/2. We want to compute f(g(x)). f(g(x))=2(…)-1=2((x/2)+1/2)-1àx+1-1=x. g(f(x))=1/2(…)+1/2àg(f(x))=x-1/2+1/2=x.
So, the important problem is g=f-1 àf(g(x))=f(f^(-1)(x))=x and g(f(x))=f^(-1)(f(x))=x.
For another example, take y=(x-1)/(x+2). This is the line with a vertical asymptot at the line x=2 and x=1. The x-intercept is equal to 1, so the point is (1,0). The y-intercept is equal to (0,-1/2). And otherwise the function graph is a hyperbola in a second and fourth quadrant and respect to its asymptoth.
Take a look at the equation of y=(x-1)/(x+2). Multiply both side with (x+2), we get y(x+2)=(x-1) àyx+2y=x-1 àyx-x=-1-2y à (y-1)x=-1-2y àx=(-1-2y)/(y-1). Now, interchange the x to y and y to xà y=(-1-2x)/(x-1). When x=0, we get y=-1 and when y=0 we get -1-2x=0 à -2x=-1 àx= -1/2.
With the vertical asymptot at x=1 and horizontal asymptot at y=-2, again we get the hyperbola. If we look carefully, we will see two functions that reflect each other.
Minggu, 14 Desember 2008
Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term (higher) arithmetic when referring to number theory, but this should not be confused with elementary arithmetic.
a. History
The prehistory of arithmetic is limited to a very small number of small artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 18,000 and 20,000 BC.
It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic by 1800 BC, although historians can only guess at the methods utilized to generate the arithmetical results - as shown, for instance, in the clay tablet Plimpton 322, which appears to be a list of Pythagorean triples, but with no workings to show how the list was originally produced. Likewise, the EgyptianRhind Mathematical Papyrus (dating from c. 1650 BC, though evidently a copy of an older text from c. 1850 BC) shows evidence of addition, subtraction, multiplication, and division being used within a unit fraction system.
Nicomachus (c. AD60 - c. AD120) summarised the philosophical Pythagorean approach to numbers, and their relationships to each other, in his Introduction to Arithmetic. At this time, basic arithmetical operations were highly complicated affairs; it was the method known as the "Method of the Indians" (Latin "Modus Indorum") that became the arithmetic that we know today. Indian arithmetic was much simpler than Greek arithmetic due to the simplicity of the Indian number system, which had a zero and place-value notation. The 7th centurySyriac bishop Severus Sebhokt mentioned this method with admiration, stating however that the Method of the Indians was beyond description. The Arabs learned this new method and called it "Hesab" or "Hindu Science". Fibonacci (also known as Leonardo of Pisa) introduced the "Method of the Indians" to Europe in 1202. In his book "Liber Abaci", Fibonacci says that, compared with this new method, all other methods had been mistakes. In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.
Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Hindu-Arabic numerals and decimal place notation for numbers. Hindu-Arabic numeral based arithmetic was developed by the great Indian mathematicians Aryabhatta, Brahmagupta and Bhāskara I. Aryabhatta tried different place value notations and Brahmagupta added zero to the Indian number system. Brahmagupta developed modern multiplication, division, addition and subtraction based on Hindu-Arabic numerals. Although it is now considered elementary, its simplicity is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician Archimedes devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. The flourishing of algebra in the medievalIslamic world and in RenaissanceEurope was an outgrowth of the enormous simplification of computation through decimal notation.
b. Decimal arithmetic
Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,...,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: for example, 507.36 denotes 5 hundreds (10²), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10-1) plus 6 hundredths (10-2). An essential part of this notation (and a major stumbling block in achieving it) was conceiving of zero as a number comparable to the other basic digits.
Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations {...,10²,10,1,10-1,...} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.
where a and b are real numbers, and i is the imaginary number which has a characteristic i2 = -1. The real number a is called the real part of the complex number and b is the imaginary part. The complex number is equal to the real number a if the value of b is 0.
For example, 3 + 2i is a complex number with the real part 3 and the imaginary part 2.
Complex number can be added, subtracted, multiplied, and divided as the real number, but it has some interesting additional characteristics. Such as, polynomial algebraic equation has a solution of the complex number, unlike the real number which has a half only.
In some diciplines (such as electrical engineering, where i is a symbol for current), the imaginary unit are written as a + bj.
a. Notation and Operation
The set of the complex number is denoted by C, or . The real number, R, can be called as a subset of C by considering every real number as a complex number :
a = a + 0i.
Complex number are added, suntracted, multiplied, and divided with the laws of algebra such as associative, commutative, and distributive, with the equation i2 = -1 :
(a + bi) + (c + di) = (a+c) + (b+d)i
(a + bi) − (c + di) = (a−c) + (b−d)i
(a + bi)(c + di) = ac + bci + adi + bd i2 = (ac−bd) + (bc+ad)i
The division of the complex number can be defined. So that, the set of the complex number can build a mathematics plane that different with the real number as an enclosed algebra.
In mathematics, the adjective of the "complex" has the meaning that the complex number is used as the based of usednumber theory. For example, complex analysis, complex matrix, complex polynomial, and complex Lie algebra.
b. Definition
The formal definotion of the complex number is a pair of the real number (a, b) with the operation of:
(a, b) + (c, d) = (a + c, b +d)
(a, b) . (c, d) = (ac - bd, bc + ad)
The complex numbers forms the set of complex number denoted by C from above definition.
Since complex number a + bi is an uniquely specified by the real number pair (a, b), the complex number has a relation of of on-on-one corespondence with the points on a plane called the complex plane.
The real number a can be called by the complex number (a, 0). Through this way, the set of the real number R be the subset of the complex number set C. | 677.169 | 1 |
Maths Made Easy
Description
Covering all the basic mathematical concepts and systems required by National Curriculum Key Stage 2, this book contains useful tips, hints, secrets and pictorial aids, and should also be suitable any young person between 8-18 years who needs extra assistance with basic mathematical principles.show more | 677.169 | 1 |
Probability Theory: A Concise Course (Dover Books on Mathematics)
Dover books are great, they cost less + they tend to have nice pencil-like images which anyone who likes mathematics would love. This book covers a lot in its concise set of pages. Do note, that you need some basic grasp of mathematics/statistics in general to gain from this book. But well worth the buy.
Ramanujan's Lost Notebook
This book covers findings of the genius Ramanujan's lost notebook. Most of his findings were scribbled in notebooks without proof. Mathematicians later recovered this book, provided proofs and mapped his adhoc findings to later discoveries. Own this if you are fascinated by his workGood question. In general, I want to recommend only books that are generally readable. A way I flag a book as unreadable for the beginner-statistician/engineer is a book that is filled with lemma's & theorems and offer no "plain-english" explanation of concepts. But if you are really into it, I would recommend getting into text books that deal with quantum physics which requires a fairly daunting amount of probabilistic math or subscribe to and read up the papers.
However, there is one I would definitely recommend, it is on machine learning and gives a good mix of plain-speak and technical math.
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Its a decent enough entry-level tutorial to cover the basics. One thing I'd recommend for sure is to maybe stress the fact that subtraction of vectors is non-commutative -- you essentially say this, but I think stressing this in mathimatical parlance is important.
Another thing you may want to consider is that it most certainly *is* possible to multiply vectors -- which you say specifically cannot be done. For all vectors, you have the dot-product, and for vectors of dimensions 3 and 7 you have the cross product -- both forms of multiplication. It's further worth mentioning that, while the cross product does not officially exist in other dimensions, its analogue may have a meaningful interpretation. For example, in 2 dimensions, the analogue of the crossYou can also do a straight-across, or 'piece-wise' multiplication between two vectors of any dimension, which results in a non-uniform scaling of the target vector. Geometrically, I don't think its terribly useful in practice, but it does have usees in other domains -- remember that vectors are used in many, many applications, not just ones which are more-or-less geometric in nature.
[Edited by - Ravyne on November 12, 2009 2:20:43 AM]
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Another thing you may want to consider is that it most certainly *is* possible to multiply vectors -- which you say specifically cannot be done. For all vectors, you have the cross-product, and for vectors of dimensions 3 and 7 you have the dot product -- both forms of multiplication. It's further worth mentioning that, while the dot product does not officially exist in other dimensions, its analogue may have a meaningful interpretation. For example, in 2 dimensions, the analogue of the dotI think you swapped dot for cross and vice versa in a couple of places. (It's clear what you meant, but it might be confusing to those who aren't already familiar with the material :)
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Thank you very much for your replies. I have updated the tutorial based on the suggestions provided. That was a good call on the vector multiplication. I never really thought about how the dot product is really just a component wise multiplication of 2 vectors. Amazing how that slipped my mind... | 677.169 | 1 |
This sequence of six books is meant for almost all of
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A proof of the technology of numbers and house, with basic demonstrations of mathematics, algebra, trigonometry, arithmetic in tune and paintings, and together with a few tips and video games utilizing numbers.
In May 1430, Joan was captured by the English and accused of witchcraft. • Joan insisted that her visions came from God, so a tribunal of French clergy condemned her as a heretic. She was burned at the stake in Rouen on 30 May 1431. II \W/5 sou! c'1l115, ill /429. She thell took the Dnllplllll to Rlleill/s, to ve crolvlled C/1I1rles Vll. Knights ••••••• • • Knights were the elite fighting men of the Middle Ages, highly trained for combat both on horseback and on foot. • Knights always wore armour.
Inca builders cut and fitted huge stones with astonishing precision to create massive buildings. • The royal palace had a garden full of life-like corn stalks, animals and birds made of solid gold. FASCINATING FACT The Inca capital was called Cuzco, which means 'navel' because it was the centre of their world. Christopher Columbus ~ The venutiful sllOres of the Bnhnll/(/s lvere provnvly Ihose first spot/erf vy Colulllvus all his voynge westwnrrf. 58 • Christopher Columbus ( 14511596) was the Genoese sailor who crossed the Atlantic and 'discovered' North and South America for Europe.
He set up a colony on Hispaniola, but i't was a disaster. Spaniards complained of his harsh rule and many Indians died from cruelty and disease. Columbus went back to Spain in chains. • Columbus was pardoned, and began a fOLirth voyage in 1502. He died off Panama, still thinking it was India. - - - 4th voyage 1502-04 ... CollIJ/lIJ/I~'sfir~/ n/ld los/ l'o)'ngcs 59 The Ref'ormation •• • In the early 15005) many people were starting to question the teachings of the Catholic Church. They were angered by the power of church leaders and the life of idleness that many monks seemed to lead. | 677.169 | 1 |
An interactive log for students and parents in my Pre-Cal 20S class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.
Wednesday, October 12, 2005
Blogging For The Test
One particular class I remember is the one when we stood on the tables. That was a bunch of laughs, huh? That class, we learned about the two right angle triangles.
My process in the class is processing. I am beggining to understand more. But I don't seem to remember how to use SOCSOATOA or how to do trigonomtry. That is what I am struck on, the trigonomtry problems.
Learning about parallel and perpendicular lines was "cool". They are so simple. All you have to know is parallel lines, the slope is always the same; perpendicular lines are the negative recipocal (the denominator and numerator are flipped).
I had problem use the different forms. I did not know when and how to use the different forms. I was very glad when Mr. K explain slowly that depending on what information is given how and what form to use. It was a reliefing feeling, I guess.
No offences, but my friend that goes to university says that calculus is not use much in life unless you are an arcitect or doing somethings with computers. And I agree, calculas is not used a lot in life, but it is used a lot the class. Like, in my geography, we had to find points and graph them and in my science class, we also had to find points and graph them. | 677.169 | 1 |
Organized for easy reference and crucial practice, coverage of all the essential topics presented as 500 AP-style questions with detailed answer explanations 5 Steps to a 5: 500 AP Calculus AB/BC | 677.169 | 1 |
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Discrete Mathematics Advice
Showing 1 to 1 of 1
This course, while extremely difficult, made you think critically and use abstract construct to solve modern problems across multiple disciplines. The challenges were made simpler with the mentorship of the professor and the aid of the lab assistant.
Course highlights:
This course presents an overview of mathematical analysis techniques. I learned to apply logic and theory to not just algebraic equations, but understood companies developed or engineers solutions in modern day appliances. I found the concepts taught in this course applied to engineering, computer science, and business.
Hours per week:
12+ hours
Advice for students:
Attend the webinar sessions put on by the TA, who helps breakdown the concepts and the homework problems. Definitely the best way to learn and get an A in the course. | 677.169 | 1 |
Lectures on Advanced Mathematical Methods for Physicists
This book surveys Topology and Differential Geometry and also, Lie Groups and Algebras, and their Representations. The former topic is indispensable to students of gravitation and related areas of modern physics, including string theory. The latter has applications in gauge theory and particle physics, integrable systems and nuclear physics, among many others. The style of presentation is such that the mathematical statements are succinct and precise, but skip involved proofs that are not of primary importance to the physics reader.
Part I provides a simple introduction to basic topology, followed by a survey of homotopy. Calculus of differentiable manifolds is developed, after which a Riemannian metric is introduced along with the key concepts of connections and curvature. The final chapters lay out the basic notions of homology and De Rham cohomology as well as fibre bundles, particularly tangent and cotangent bundles.
Part II starts with a review of group theory followed by the basics of representation theory. A thorough description of Lie groups and algebras is presented with their structure constants and linear representations. Next, root systems and their classifications are detailed, concluding with the description of representations of simple Lie algebras emphasizing on spinor representations of orthogonal and pseudo-orthogonal groups. | 677.169 | 1 |
TECHNIQUES OF INTEGRATION 7.6 Integration Using Tables and Computer Algebra Systems In this section, we will learn: How to use tables and computer algebra systems in integrating functions that have elementary antiderivatives.
However, you should bear in mind that even the most powerful computer algebra systems (CAS) can't find explicit formulas for: The antiderivatives of functions like e x 2 The other functions at the end of Section 7.5
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TABLES OF INTEGRALS Remember, integrals do not often occur in exactly the form listed in a table. Usually, we need to use substitution or algebraic manipulation to transform a given integral into one of the forms in the table.
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Master Math
Trigonometry
A trigonometry guide that explains the subject matter in a user-friendly manner. It provides step-by-step solutions for things you need to know about the subject including trigonometric functions, inverse trigonometric functions, trigonometric identities, equations and inequalities, vectors, polar coordinates, and more. | 677.169 | 1 |
Description: After completing this course, students should have developed a clear understanding of the fundamental concepts of single variable calculus and a range of skills allowing them to work effectively with the concepts.
The basic concepts are: 1. Derivatives as rates of change computed as a limit of ratios 2. Integrals as a 'sum' computed as a limit of Riemann sums
After completing this course, students should demonstrate competency in the following skills: Use both the limit definition and rules of differentiation to differentiate functions. Sketch the graph of a function using asymptotes, critical points, the derivative test for increasing/decreasing functions, and concavity. Apply differentiation to solve applied max/min problems. Apply differentiation to solve related rates problems. Evaluate integrals both by using Riemann sums and by using the Fundamental Theorem of Calculus. Apply integration to compute arc lengths, volumes of revolution and surface areas of revolution. Evaluate integrals using advanced techniques of integration, such as inverse substitution, partial fractions, and integration by parts. Use L'Hospital's rule to evaluate certain indefinite forms. Determine convergence/divergence of improper integrals and evaluate convergent improper integrals. Determine the convergence/divergence of an infinite series and find the Taylor series expansion of a function near a point.
Resources: OpenCourseware from MIT, UC Berkeley, Stanford along with many of the World's finest University's. | 677.169 | 1 |
Math 20-3 2016 Course Outline SEREDA
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St. Francis Xavier Mathematics 20-3 ~ Course Outline Ms Sereda: Feb 2016 –June 2016 OVERVIEW Mathematics 20-3 will provide students with the mathematical understandings and critical thinking skills identified for entry into the majority of trades and for direct entry into the work force. COURSE MATERIALS STUDENT RESOURCES SUPPLIES MathWorks 11 – Mandatory! Graphing Calculator Binder/Paper/Graph Paper Pen/Pencil/Eraser/Ruler etc. PRE-REQUISITE Math 10-3 with a 50% or higher (preferably 60%+) COURSE OVERVIEW The units to be studied and the approximate number of classes spent on each unit are shown below: Chapter Topic Time (classes) Completion date (approx.) 2 Graphical Representations Ongoing March 18th 6 Financial Services 12 February 24 7 Personal Budgets 11 March 15th Midterm 3 (review) March 22 3 Surface Area 7 April 12 3 Volume and Capacity 6 April 22 1 Slope and Rate of Change 8 May 4 5 Scale Representations 11 May 19 4 Solving Right Triangles 6 June 8 Final Exam + review 2 (review) Jun. 14 or 15 Course outlines and dates are subject to review based on the professional discretion of the teacher and the needs of the student. COURSE EVALUATION Your final course mark will be calculated according to the following scale: Student Assessment Quizzes (Pop), Assignments 15% Projects 20% Chapter Exams 25% Midterm Exam (Chapters 2, 6, 7) 15% Final Exam (Chapters 3, 1, 5, 4) 25% St. Francis Xavier Mathematics 20-3 ~ Course Outline Ms Sereda: Feb 2016 –June 2016 Note: The Final exam is an in-class exam and MUST be written in order to pass the course COURSE EVALUATION METHODS Homework: After each class there will be a homework assignment found from the textbook. It is highly recommended and expected students attempt these homework assignments as practice in this subject truly makes perfect! Workbooks need to be brought to class everyday as well as on the day of Chapter Exams. Quizzes: There will be 1- reasons will be accepted (please see student handbook). Students may NOT miss another class to write a missed exam. Please notify the teacher prior to extended absences if possible. Note: A pattern of repeated absences from exams/lessons will require a parent/student/teacher meeting. 4. Important A wide range of assessment information is used in the development of a students' final grade. At St. Francis Xavier, individualized assessment provide specific information regarding student progress and overall performance in class. Student assessment may vary from student to student to adapt for differences in student needs, learning styles, preferences and paces. It should also be noted that not all assignments are used to determine the final grade. Please feel free to contact me if you have any questions or would like to share any concerns. I can be reached in several ways. If you would like to talk to me, please call the school and leave a phone number that I can reach you at and I will get back to you as soon as I can. The school phone number is: 780 489-2571 You can also e-mail me at the following address: tara.sereda@ecsd.net I would be happy to hear from you at any time. Please read the course syllabus that your son or daughter has brought home and return the final page of the course outline. Sincerely, Tara Sereda St. Francis Xavier Mathematics 20-3 ~ Course Outline Ms Sereda: Feb 2016 –June 2016 I have read and understood the course syllabus for my son/daughter's Math 20- 3 class: Student Name: _________________________ Course: Math 20-3 SEREDA Student Signature: _______________________ Parent Name: ________________________ __________________________ (please print) first name last name Parent Signature: ________________________ Parent e-mail #1: Parent e-mail #2 (optional; please include name of parent below e-mail): - | 677.169 | 1 |
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...Statisticians are in great demand and if the subject appeals to you, you should consider GG13. Although you can do some Statistics modules in G100, you will do more in GG13. The first year is similar to G100 except that the module in Applied Mathematics is not compulsory. Over the final two years... Learn about: Skills and Training, GCSE Mathematics, Basic IT training...
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...This course will provide students with a greater knowledge and understanding of Advanced Mathematics and help to enhance your current skills. It is the ideal course for adults or young people who would like to develop their skills. The Advanced Mathematical Skills programme is broken... Learn about: Skills and Training...
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Real-Life Math: Everyday Use of Mathematical Concepts
"What does this have to do with real life?" is a question that plagues mathematics teachers across America, as students are confronted with abstract topics in their high school mathematics courses. The National Council of Teachers of Mathematics emphasizes the importance of making real world connections in teaching mathematics so that learning new content is meaningful to students. And in meeting NCTM national standards, this invaluable book provides many insights into the many connections between mathematics applications and the real world. Nearly 50 math concepts are presented with multiple examples of how each is applied in everyday environments, such as the workplace, nature, science, sports, and even parking. From logarithms to matrices to complex numbers, concepts are discussed for a variety of mathematics courses, including:
algebra geometry trigonometry analysis probability statistics calculus In one entry, for example, the authors show how angles are used in determining the spaces of a parking lot. When describing exponential growth, the authors demonstrate how interest on a loan or credit card increases over time. The concept of equations is described in a variety of ways, including how business managers estimate how many hours it takes a certain number of employees to complete a task, as well as how a to compute a quarterback's passing rating. Websites listed at the end of each entry provide additional examples of everyday math for both students and teachers. | 677.169 | 1 |
Abstract Algebra [Paperback]
The book preserved the emphasis on providing a large number of examples and on helping students learn how to write proofs. The presentation of the sections given at a higher level. Unusual features, for a book is still relatively short, are the inclusion of full proofs of both directions of Gauss theorem on constructible regular polygons and galois theorem on solvability by radicals, a Galoistheoretic proof of the Fundamental Theorem of Algebra, and a proof the Primitive Element Theorem.
First semester course should probably include the material of Sections 0-13, and some of the material on rings in Section 16 and the following sections, Sections 14 and 15 allow the inclusion of some deeper result on groups. In second semester it should be possible to cover the whole book, possibly omitting Section 21.
The changes include the simplification of some points in the edition of some new exercise, and the updating of some historical material. All the topics are given in step by step method with simple language to understand the concept easily. This book is intended for use in a junior-senior level course in abstract algebra. The students who used the book for the five sections as text and pointed out to me parts of the presentation that needed clarification. | 677.169 | 1 |
I have a test tomorrow afternoon. But I'm stuck with problems based on free ged printable prep. I'm having problems understanding like denominators and graphing inequalities because I just can't seem to figure out a way to solve problems based on them. I called my friends and I tried on the internet, but neither of those activities helped. I'm still trying but the time is short and I can't seem to get things moving . Can somebody please show me the way? I really need some help from you guys for tomorrows assignment. Please do reply.
I find these routine queries on almost every forum I visit. Please don't misunderstand me. It's just as we advance to college, things change suddenly . Studies become complex all of a sudden. As a result, students encounter trouble in completing their homework. free ged printable prep in itself is a quite challenging subject. There is a program named as Algebrator which can help you in this situation.
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Algebrator is the program that I have used through several algebra classes - Basic Math, Intermediate algebra and Algebra 1. It is a truly a great piece of algebra software. I remember of going through difficulties with graphing function, roots and adding fractions. I would simply type in a problem from the workbook , click on Solve – and step by step solution to my math homework. I highly recommend the program.
Hi All, Based on your comments , I bought the Algebrator to get myself educated with the fundamental theory of Algebra 2. The explanations on function definition and adding functions were not only graspable but made the whole topic pretty interesting. Thanks a million for all of you who pointed me to have a look at the Algebrator! | 677.169 | 1 |
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Review of Math Essentials' No-Nonsense Algebra
Homeschool math at the high school level can be intimidating so I'd like to introduce you to No-Nonsense Algebra from Math Essentials. Finding a math program that our 14-year-old daughter likes and understands can be challenging. But with the included videos for the lessons taught by Richard W. Fisher, our daughter has found this program to be understandable.
Our daughter is our 4th and last child and she is definitely more gifted in the area of art than math. So we have tried a variety of math programs through the years. Like most parents, we have found some work and some do not. So when I saw the chance for our family to review No-Nonsense Algebra, I was glad to get it for our daughter.
This is a soft cover book that is meant to be used as a textbook, and the students need to do their work in a notebook. Since it is not consumable, it would work well for families with more than one child. As I have already stated, our daughter is not a math whiz kid. She loves art a lot more. So in her own words, she stated, "Some of the instructions and problems are worded in 'mathease', which is a language our daughter feels she does not speak. However, the videos provided our daughter with clear and concise instructions that made the lessons understandable for her.
I feel like the videos are well-executed, easy to hear and easy to understand. The instructions really helped our daughter. This course covers everything from integers to the quadratic equation along with everything in between.
I felt like the lessons were reasonable in the material being covered, the examples that were given and the number of practice problems. Each lesson contains an introduction which includes helpful hints, example problems worked out showing each step for solving the problem, exercises, and several review questions.
Since I have taught high school math and realize how it builds upon itself, I love the review problems. It helps to ensure the students remember past materials. I also loved the fact the book starts off with problems involving integers. This is a necessary skill for solving most algebraic problems. If a student can not solve problems with integers properly, he or she will probably have extreme difficulty later in Algebra. Also, from my experience in teaching Algebra, the majority of mistakes in Algebra are caused by mathematical computations. So ensuring students have a firm grasp on working with integers will help with their success. The pictures below show two days of my daughter's work and answers. As I have stated, she is very artistic so her pages have to include some drawings.
I feel this book is perfect for a variety of situations. It is perfect as a stand-alone textbook for Algebra. It is also perfect for a student who just wants to brush up on certain skills. It also would be perfect for a student just wanting to review Algebra for any reason. I also like the fact Math Essentials offers a variety of other products ranging from materials for 4th/5th grade all the way through this Algebra course.
I was very pleased with this book and the videos are provided with it. This course is definitely worth taking a look at in my opinion. If you would like to read more reviews about No-Nonsense Algebra from Math Essentials, click here or on the graphic below | 677.169 | 1 |
Mathematics GCSE
Exam Board
Level
Qualification
Assessment Method
OCR
Level 2
GCSE
100% Exam Higher Calculator and Non-Calculator course. Students will have the opportunity of taking the GCSE Foundation Exam in November and if required, the following summer.
Starting in September students will study for 5 periods per week during an intensive 8 week course up to the first re-sit opportunity in early November. During this period a lot of content is covered and students will be expected to start revising within the first couple of weeks. After the November exams students will start a more extensive course, again for 5 periods per week, that will then run through to May. Once exam results are available in early January those students who have achieved a grade 4 or 5 will be allowed to leave the course, while those students who do not achieve this grade continue and take a further re-sit in June. All content covered during the year will be similar to that students have studied in previous years at Secondary School, but a particular focus is placed on making minimal mistakes and picking up understanding of topics that students have found difficult in the past. Topics covered are divided up into four sections:
Mathematics is a core subject and as such is necessary for any student considering applying for an academic course at university. Maths is valued by employers with skills in numeracy and problem solving being highly sought after by employers | 677.169 | 1 |
Moved Permanently. The document has moved here. Casio's CAS is based on the Casio's original algebra system and enhanced by our R D If you have switched from the 9850 Series to the ALGEBRA FX 2. 0 PLUS.
A graphic calculator from Casio, the Casio Algebra FX 2. 0 The Algebra FX 2. 0 versions Casio graphic calculators use a BASIClike programming language. Activity 4 Algebra I with the Casio fx9750GII 2. Press d then u(TABL) to display the table. 3. To display a corresponding yvalue for a specific | 677.169 | 1 |
Description: This is a first course on group theory but is more suitable to a third year student than a first year one. It attempts to motivate group theory with many illustrative examples such as shuffling of cards, bell ringing and permutation puzzles. As well as the usual introductory theory there's an elementary introduction to representation theory, to the Todd-Coxeter algorithm and to free groups.Galois Groups and Fundamental Groups by Leila Schneps - Cambridge University Press This book contains eight articles which focus on presenting recently developed new aspects of the theory of Galois groups and fundamental groups, avoiding classical aspects which have already been developed at length in the standard literature. (8014 views)
Combinatorial Group Theory by Charles F. Miller III - University of Melbourne Lecture notes for the subject Combinatorial Group Theory at the University of Melbourne. Contents: About groups; Free groups and presentations; Construction of new groups; Properties, embeddings and examples; Subgroup Theory; Decision Problems. (8476 views) | 677.169 | 1 |
The generalized formula, applicable on any uniform polyhedron having 2n congruent right kite faces, 4n edges & 2n+2 vertices lying on a spherical surface with a certain radius, have been derived by the author Mr H.C. Rajpoot to analyse infinite no. of the uniform polyhedrons (trapezohedrons) having congruent right kite faces simply by setting n=3,4,5,6,7,………………upto infinity.Julius Caesar is one of Shakespeare's greatest and most performed plays. Included in this guide: a biography of author William Shakespeare, a look at the play's context, its literary elements, detailed scene summaries, character analysis, and much more. This is the definitive guide to Julius Caesar, concise, easy to understand, and guaranteed to add to your enjoyment of this classic play.
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Tables of solid angles subtended at the vertices by all 5 platonic solids (regular polyhedrons) & all 13 Archimedean solids (uniform polyhedrons) calculated by the author using standard formula of solid angle & formula of tetrahedron. These are the standard values of solid angles which are useful for the analysis of platonic solids & Archimedean solids.
In this book, the fundamental principles, definitions and formulas necessary for the in-depth understanding of Electric Circuits are introduced, along with characteristic examples and problems, accompanied by their solution. It is an excellent introductory textbook for students of Electrical Engineering and related disciplines.
15 Tips that are tested and Mom approved for assisting you in dealing with School Districts within the IEP Meetings! As a parent, it is imperative you go to the meeting prepared and ready to have to fight for your rights! This book lists the Top 15 Tips you need to know before going to the IEP Meeting!
This is the teaching resource book that can be used with Astro is Down in the Dumps. It is only available as a PDF on Smashwords so that it can be printed and used in the teaching environment. There is also a weblink and a password at the back of the book so you can download a PDF at anytime.
This book examines the economics of the postal sector through three lenses: snapshot and trends, models, and opportunities. In the years to come, the Universal Postal Union plans to develop its role as a knowledge centre for the postal sector from these perspectives.
Teacher Edition includes complete student text, review questions, vocabulary, and answer keys. The Conflict of the Ages is a Multi-Part exploration of History, Science and ancient Literature. This first installment covers the concepts of God, time, Creation, physics, cosmology, and specifics about each day of Creation.
Teacher edition includes full student text with review questions, vocabulary, and answer keys. God What does "their eyes were opened" really mean?
The worldwide Flood is one of the most discounted records in the Scriptures. Yet it is supported around the world by historical accounts. Take a look at feasibility studies on the safety and the stocking of the Ark.
The Geologic Column ought to prove that fossils reveal the age of the earth. They show progression from simple to complex organisms over millions of years. But do they?
Generalized formula of a tetrahedron have been derived by the author Mr H.C. Rajpoot by using HCR's Inverse cosine formula & HCR's Theory of Polygon. These formula are very practical & simple to apply in case of any tetrahedron to calculate the internal (dihedral) angles between the consecutive lateral faces meeting at any of four vertices & the solid angle subtended by it (tetrahedron) at vertex
Macbeth is one of Shakespeare's greatest and darkest plays. Included in this guide: a biography of author William Shakespeare, a look at the play's context, its literary elements, detailed scene summaries, character analysis, and much more. This is the definitive guide to Macbeth, concise, easy to understand, and guaranteed to add to your enjoyment of this classic play. | 677.169 | 1 |
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It is an algebraic system like a ring where elements can be multiplied by real or complex number including other elements too. A special system of notation is adapted to study special system of relationship which is termed as algebra of classes. | 677.169 | 1 |
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Math Final Review Dec 10, 2010
Transcription
1 Math Final Review Dec 10, 2010 General Points: Date and time: Monday, December 13, 10:30pm 12:30pm Exam topics: Chapters 1 4, 5.1, 5.2, 6.1, 6.2, 6.4 There is just one fundamental way to prepare for the final exam: understand the material! To review for the exam, I suggest studying all the midterm exams and the homework problems. Solutions to the midterm exams and selected problems from the homework can be found on the course web page. You will be allowed to prepare and use an 8.5 inch by 11 inch piece of paper with your notes during the exam. You will also be allowed to use a calculator. You will answer questions on the exam itself. There will be a mix of true/false and problems on the exam. To receive full credit on the problems, all work must be shown and correct. When you receive the exam, relax and proceed deliberately. If you don t know how to do a problem, skip it and return to it later. Accuracy is paramount, speed is useless! The exam is not a contest to see who can finish the fastest. If you finish early, be content that you now have time to double and triple check your answers. Chapter 1: 1.1: Vectors and linear combinations 1. Know both the algebraic and geometric view of taking linear combinations of vectors. What do all linear combinations of linearly independent vectors give you? 2. Parallelogram rule 1.2: Lengths and dot products 1.3: Matrices 1. Formula for dot product between two vectors. 2. Geometric meaning of the dot product between two vectors; cosine forumla. 3. How to find the length (norm) of a vector in terms of dot products. 4. Definition of a unit vector and how to compute it. 5. Know how to apply Schwarz inequality: v w v w. 6. Know how to apply Triangle inequality: v + w v + w 1. Two views of matrix-vector multiplication Ax: 1) linear combination of columns of A; 2) Dot product of rows of A with x. 2.1: Vectors and linear equations 1. Geometric view of solving linear system of equations Ax = b. (a) Row view: intersecting planes (b) Column view: adding vectors together to produce b. 1
2 2 2.2: Elimination idea 1. Know how to put a linear system Ax = b into upper triangular form so that the system can be easily solved by back substitution. 2. Know the three things that can go wrong in elimination and what they mean in terms of solving the linear system of equations. (a) A pivot=0. (b) Reduced augmented system has a row of all zeros. (c) Reduced augmented system has a row of all zeros except the last column. 2.3: Elimination using matrices 1. Know how to do elimination using matrix products on the augmented system. One can reduce the system to upper triangular form using a sequence of multiplications of the system by lower triangular matrices. 2.4: Rules for matrix operations 1. Know properties for matrix multiplication: associative law, left and right distributive law, identity law. 2. What about a commutative law? 3. Two views of matrix multiplication. 4. Know how many numbers must be multiplied when doing matrix-matrix multiplication and how to intelligently multiply several matrices together. 2.5: Inverse matrices 1. Definition of the inverse of a square matrix. 2. A 1 exists if and only if using elimination to solve Ax = b produces n non-zero pivots. 3. If the inverse of A exists then it is unique. 4. If A is invertible then the solution to Ax = b is x = A 1 b. 5. If A is invertible, then the only solution to Ax = 0 is x = Know how to compute the inverse of a 2-by-2 matrix. 7. Know how to find the inverse of a diagonal matrix. 8. Rule for splitting up the inverse of a product. 2.6: LU factorization 1. Definitely know how to compute the LU factorization of a matrix. 2. Know how to solve a linear system of equations using LU factorization. 2.7: Transposes and Permuations 1. Properties of the transpose of a matrix. 2. Definition of a symmetric matrix. 3. Know how to compute inner and outer products using the transpose. 3.1: Vector spaces 1. Definition of a vector space and the 8 properties a vector space must satisfy (see page 127). 2. Definition of a subspace of a vector space. 3. Definition of the column space of a matrix and how it relates to Ax = b having a solution. 3.2: Nullspace 1. Definition of the nullspace of a matrix, N(A) 2. Know how to find N(A) using rref.
3 3 3.3: Rank 3. If A is invertible what is N(A). 1. Definition of the rank(a) in terms of pivots, independent rows and columns of A, and the dimension of C(A) and C(A T ). 2. What is the rank of xx T? 3.4: Complete solution to Ax = b 1. Know how to find the complete solution of Ax = b. 2. Definition of matrices with full column or full row rank. 3. Definitely know the relationship between the number of possible solutions to Ax = b and the rank and size of A (4 possibilities). 3.5: Linear independence, basis, and dimension 1. When is a set of vectors linearly independent? 2. What does the span of a set of vectors mean? 3. What do C(A) and C(A T ) mean in terms of the span of some vectors? 4. Know the definition of a basis for a vector space. 5. Know how to find a basis for C(A), C(A T ), and N(A). 6. How is the dimension of a subspace defined in terms of a basis? 3.6: Dimension of the four fundamental subspaces 1. Definition of the four fundamental subspaces of a matrix and the sizes of the vectors in these spaces. 2. Definitely know The Big Picture in Figure 3.5, p Definitely know part I of the fundamental theorem of linear algebra. 4. Know how to find a basis for N(A T ). 4.1: Orthogonality of the four fundamental subspaces 1. What does it mean for two vectors to be orthogonal? 2. What is the geometric picture of two orthogonal vectors? 3. Definition of a orthogonal subspace. 4. Definition of the orthogonal complement of a subspace and how to find it. 5. Part II of the fundamental theorem. 6. If A is m-by-n then any vector in R n can be written as x = x r + x n, where x r is in C(A T ) and x n is in N(A). 7. Definitely know the picture displayed in Figure 4.3 showing the true action of a matrix A on a vector x. Where does Ax r go? Where does Ax n go? Can you reproduce this picture if asked? 8. If you have n independent vectors in R n, what is the span of these vectors equal to? 4.2: Projections 1. Geometric picture of the projection of a vector onto a subspace (in two dimensions). 2. How to project onto a line. 3. If V is a subpace and p is the projection of b onto V then b p is?? to any vector from V. 4. Know how to project onto a subspace V. 5. What property guarantees A T A is invertible? 6. You should definitely know how to project a vector onto a subspace. For example, a really good question is the extra credit question from midterm 3. This involves all sorts of things we have learned and the key to solving it is to use projections!
4 4 4.3: Least squares 1. When does Ax = b have a solution? 2. There will definitely be a problem on computing the least squares solution to an overdetermined system. 4.4: Gram-schmidt and orthogonal bases 1. Definition of a set of orthogonal and orthonormal vectors. 2. Nice property of matrices with orthonormal columns (in the case of square matrices we call these orthogonal matrices). 3. Know how to do projections with orthogonal bases. 4. Given a set of vectors, know how to use Gram-Schmidt to convert the set to an orthonormal set that spans the same space. 5. Know how to compute the QR decomposition of a matrix. 6. Know the technique for solving least squares problems with QR decomposition. 5.1: Determinants 1. Know formula for calculating the determinant of a 2-by-2 matrix. 2. Know the 10 properties regarding the determinant given on pp and how to apply them. 5.2: Permutations and cofactors 1. Know how to find the determinant of A from the P T LU decomposition of A. 2. Know how to compute the determinant of a matrix using cofactors. 6.1: Eigenvalues and eigenvectors 1. Know how to find the eigenvalues and eigenvectors of a matrix using the characteristic equation and rref. 2. Know the geometric meaning of an eigenvalue, eigenvector of a matrix. 3. Know the relationship between the determinant of a matrix and its eigenvalues. 4. Know the relationship between the trace of a matrix and its eigenvalues. 5. Relationship between the eigenvalues and eigenvectors of A and A 2, A 3,..., A k (k > 1). 6. Relationship between inverse of A and its eigenvalues and eigenvectors. 6.2: Diagonalizing a matrix 1. Know the theorem regarding the diagonalization of a matrix. What are the assumptions? 2. Know how to diagonalize a matrix. 3. How can the powers of A be computed easily from the eigenvector diagonalization? 4. Properties of diagonalization (a) If A has n independent eigenvalues then the eigenvectors are independent. So A can be diagonalized. (b) The eigenvectors of S come in the same order as the eigenvalues of Λ. (c) If x is an eigenvector then so is cx where c is any non-zero constant. (d) If A does not have n linearly independent eigenvectors then it cannot be diagonalized. (e) If A has a zero eigenvalue then A is not invertible. It may still be diagonalizable (if its n eigenvectors are independent). 5. If you know the eigenvalues and eigenvectors of A then what are the eigenvalues and eigenvectors of A + ci, where c is a constant and I is the identity matrix. 6. Pitfalls:
5 5 (a) If λ is an eigenvalue of A and β is an eigenvalue of B then λβ is not generally an eigenvalue of AB. (b) Also λ + β is not generally an eigenvalue of A + B. 6.4: Symmetric matrices 1. A symmetric matrix has only real eigenvalues. 2. A symmetric matrix has orthonormal eigenvectors. 3. If A is symmetric then it can be diagonalized as A = QΛQ T, where Q is orthogonal and contains the eigenvectors of A in its columns. 4. Every symmetric matrix can be diagonalized. See the conceptual question review starting on page 552 of the book.MATH 54 FINAL EXAM STUDY GUIDE PEYAM RYAN TABRIZIAN This is the study guide for the final exam! It says what it does: to guide you with your studying for the exam! The terms in boldface are more important
MA 5 May 9, 6 Final Review This packet contains review problems for the whole course, including all the problems from the previous reviews. We also suggest below problems from the textbook for chapters
POL502: Linear Algebra Kosuke Imai Department of Politics, Princeton University December 12, 2005 1 Matrix and System of Linear Equations Definition 1 A m n matrix A is a rectangular array of numbers with
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
Math 480 Diagonalization and the Singular Value Decomposition These notes cover diagonalization and the Singular Value Decomposition. 1. Diagonalization. Recall that a diagonal matrix is a square matrixPractice 18.06 Final Questions with Solutions 17th December 2007 Notes on the practice questions The final exam will be on Thursday, Dec. 20, from 9am to 12noon at the Johnson Track, and will most likely
Example Linear Algebra Competency Test Solutions The 40 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter
Math 22 Final Exam. (36 points) Determine if the following statements are true or false. In each case give either a short justification or example (as appropriate) to justify your conclusion. T F (a) IfSection. 2. Suppose the multiplication cx is defined to produce (cx, ) instead of (cx, cx 2 ), which of the eight conditions are not satisfied?, 4, 5. Which of the following subsets of R are actually subspaces?Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible
Exam in Linear Algebra First Year at The TEK-NAT Faculty and Health Faculty January 8th,, 9- It is allowed to use books, notes, photocopies etc It is not allowed to use any electronic devices such as pocket
Definition: A vector space V is a non-empty set of objects, called vectors, on which the operations addition and scalar multiplication have been defined. The operations are subject to ten axioms: For any
More Linear Algebra Study Problems The final exam will cover chapters -3 except chapter. About half of the exam will cover the material up to chapter 8 and half will cover the material in chapters 9-3.
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topicsNote: A typo was corrected in the statement of computational problem #19. 1 True/False Examples True or false: Answers in blue. Justification is given unless the result is a direct statement of a theoremMATH 310, REVIEW SHEET 1 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,
Solution Set 8 186 Fall 11 1 What are the possible eigenvalues of a projection matrix? (Hint: if P 2 P and v is an eigenvector look at P 2 v and P v) Show that the values you give are all possible SolutionSolutions to Homework 8. Find an orthonormal basis for the plane 4x x + x = in R. Answer: First, we choose two linearly independent vectors that are in the plane. These vectors from a basis for the plane,
LECTURE Linear Algebra and Matrices Before embarking on a study of systems of differential equations we will first review, very quickly, some fundamental objects and operations in linear algebra.. Matrices
Math 5 Exam # Practice Problem Solutions For each of the following statements, say whether it is true or false If the statement is true, prove it If false, give a counterexample (a) If A is a matrix such
GRE math study group Linear algebra examples D Joyce, Fall 20 Linear algebra is one of the topics covered by the GRE test in mathematics. Here are the questions relating to linear algebra on the sample | 677.169 | 1 |
Engineering Mathematics / Edition 6
Now in its eighth edition, Engineering Mathematics is an established textbook that has helped thousands of students to succeed in their exams. John Bird's approach is based on worked examples and interactive problems. Mathematical theories are explained in a straightforward manner, being supported by practical engineering examples and applications in order to ensure that readers can relate theory to practice. The extensive and thorough topic coverage makes this an ideal text for a range of Level 2 and 3 engineering courses. This title is supported by a companion website with resources for both students and lecturers, including lists of essential formulae and multiple choice tests.
Product Details
Table of Contents
Preface; Part 1: Number and Algebra; Part 2: Areas and volumes; Part 3: Trigonometry; Part 4: Graphs; Part 5: Complex numbers; Part 6: Vectors; Part 7: Statistics; Part 8: Differential Calculus; Part 9: Integral calculus; Part 10: Further Number and algebra; Part 11: Differential equations; Multi-choice questions; Answers to multiple choice questions; Index | 677.169 | 1 |
The School of Mathematics
The school of mathematics cultivates a natural and stress-free environment where anyone can study, discuss, explore, and experience mathematics. No prior knowledge is assumed. Whether you are an avid student of mathematics or always shied away saying "mathematics is not for me", you are welcome. Our approach allows anyone to naively discover mathematics. All studies at the school are free of charge.
Each workshop series begins with a short background story which leads to a research question. This question can be understood by all participants, and trying to answer it reveals deep mathematical issues. In many cases, it is a question that had been studied by mathematicians before us. Without looking at any answers, we try to discover our own. We learn to develop our ideas, express them to others and work together. It may take us a few minutes and it may take more than a few hours. If we find an answer, we may look for additional answers. And we ask each other questions during our research which we then try, with the same methods, to answer.
We are working our way through the book "Algebra", by I. Gelfand. This is an elementary text&emdash; it begins with addition and multiplication, and moves on to other school topics like functions and polynomials. Most of the learning happens in the exercises, and so the experience is really one of discovery. | 677.169 | 1 |
Foundations of Mathematics
Math 281 Dave's Syllabus
Fall 2005
There's some irony to the name of this course. You've probably taken math
classes for 13 straight years and now you get to the Foundations!?!
What's all of your math knowledge built on anyway, sand? Nothing?
Actually your path through mathematics mirrors the
historical development of those same ideas. Limits and derivatives were being
used for 170 years before good definitions were developed. Various cultures
talked about a concept of infinity for centuries before Georg Cantor provided
the foundations for the mathematical study of infinity. (He proved a stunning
fact that we'll learn in this course – not only are there different sizes of
infinity, but there are actually an infinite number of sizes of infinity!)
In this semester of FOM, we'll cover about the following
topics, all of which will be vital in future math courses (and, actually, in
life):
What it means to prove something & why we prove
things.
Set theory - sets, power sets, combinations of sets.
Logical statements - the real language underlying
mathematics.
Infinity – what it means, how to define and study it.
Problem solving skills – systematic ways to get unstuck
when you're stuck on a problem.
To learn these key concepts in mathematics we'll use a
variety of classroom activities, homework, and writing assignments. Also,
you'll be expected to spend a significant amount of time reading the textbook.
Where to go for help: You have three main resources
to draw on when you need help in this class. The first and most important is
your fellow classmates. This course will be hard – at times very hard. It will
go much smoother for all of us if you start getting to know your classmates and
start studying with them outside of class early in the semester. The second is
me – my contact info and office hours appear above. I will also be around at
other times - feel free to drop by and say hi. If you can't find me, email or
call and we'll schedule an appointment that works for both of us. If an
emergency comes up and you are forced to miss class, you should drop me an
email (I check it very frequently).
Your third resource will be your TA, Josh, who excelled in
this course last year.
Assignments: There will be three different types of assignments: your
journal, written proofs, and problem solutions.
For the journal, you may choose any type of notebook/binder/daily diary.
When reading the text you should have your journal open, jotting down the
important points, answering the exploratory questions, filling in gaps, trying
to do proofs before the authors do, etc. You may also want to note any
questions you have so that you remember to ask them in class. Your journals
will be collected periodically throughout the semester.
Written Proofs will be assigned about once a week and collected in class.
You will be graded on how complete and understandable your proofs are. For
your first two proofs, you will be encouraged to revise and resubmit them.
This will give you some time to adjust to our expectations. We encourage you
to work with others to develop your proofs but the writing must be entirely
your own.
For several years we have posted Problems of the Week on the MathCS wing.
We will continue to do this and part of your grade will be based on your work
on these problems. Although you may not be able to solve each one, you must
turn in your work for each one – showing the progress you made toward
understanding and solving the problem.
Grading: We will decide as a class how much each contributes to your
grade.
Assessment
Date Percent
Midterm October
19th
Journal all
semester
Written
Proofs all semester
Problem
Solutions all semester
Class
Participation all semester
Take-home
Final Due Dec. 12th
Final
Project Due Dec. 7th
Total 100
The
mid-term will be in class – though you may start as early as 8am if you'd
like. Anyone who has an 8am class will be given an opportunity to have a
similar amount of time. The final will be a take-home exam which must be done
without consulting other people or other books. The final project is your
chance to be creative. Past projects include short films, board games, mathematical
sculptures, short stories, and musicals. | 677.169 | 1 |
Description: Generating functions, one of the most important tools in enumerative combinatorics, are a bridge between discrete mathematics and continuous analysis. Generating functions have numerous applications in mathematics, especially in: Combinatorics; Probability Theory; Statistics; Theory of Markov Chains; and Number Theory. One of the most important and relevant recent applications of combinatorics lies in the development of Internet search engines, whose incredible capabilities dazzle even the mathematically trained user.
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This classic best-seller by a well-known author introduces mathematics history to math and math education majors. Suggested essay topics and problem studies challenge students. CULTURAL CONNECTIONS sections explain the time and culture in which mathematic | 677.169 | 1 |
Web Resources
This section covers:
• Understand the importance of doing and undoing in mathematics
• Determine when a process can or cannot be undone
• Use function machines to picture and undo algorithms
• Understand that functions produce unique outputs
This sections objectives ate to find, describe, explain, and predict using patterns:
• Determine whether or not patterns in tables are uniquely described
• Distinguish between closed and recursive descriptions of patterns
• Understand that a table of data associated with a specific situation determines a unique pattern
• Understand that there are different conceptions of algebra
What are functions ? From an introduction of the basic concepts of functions to more advanced functions met in economics, engineering and the sciences, these topics provide an excellent foundation for undergraduate study. | 677.169 | 1 |
An Introduction to Measure and Probability (Textbooks in Mathematical Sciences) by John Taylor
Assuming only calculus and linear algebra, Professor Taylor introduces readers to measure theory and probability, discrete martingales, and weak convergence. This is a technically complete, self-contained and rigorous approach that helps the reader to develop basic skills in analysis and probability. | 677.169 | 1 |
This self-teaching workbook is designed especially for students who need to go back to algebra basics as preparation for starting a college-level math course. It's also a helpful review for those preparing to take standardized exams that include math testing, such as a math placement exam, the GRE or GMAT. "Forgotten Algebra" contains 32 work units, starting its review with signed numbers, symbols, and first-degree equations, and progressing to include logarithms and right triangles. Each work unit reviews basics before presenting problems and exercises that include detailed solutions designed to facilitate self-study. The book's systematic presentation of subject matter is easy to follow, and encompasses all the terminology, equations, and information that students of algebra need to master. This new edition has been expanded to include step-by-step solutions for all exercises. | 677.169 | 1 |
section: the post-emancipation period. yet what did this suggest for the Jewish neighborhood and their interactions with wider society?
This e-book develops the topic of matrices with targeted connection with differential equations and classical mechanics. it really is meant to convey to the scholar of utilized arithmetic, without past wisdom of matrices, an appreciation in their conciseness, strength and comfort in computation. labored numerical examples, lots of that are taken from aerodynamics, are incorporated.
General zero fake fake fake The Blitzer Algebra sequence combines mathematical accuracy with an attractive, pleasant, and sometimes enjoyable presentation for max attraction. Blitzer's character indicates in his writing, as he attracts readers into the cloth via correct and thought-provoking purposes.
The product of a number and 3 is 6. 73. Three times a number is equal to 8 more than twice the number. 74. Twelve divided by a number equals 13 times that number. Identify each as an expression or an equation. See Example 6. 75. 3x + 21x - 42 76. 8y - 13y + 52 77. 7t + 21t + 12 = 4 78. 9r + 31r - 42 = 2 79. x + y = 9 80. x + y - 9 A mathematical model is an equation that describes the relationship between two quantities. 212x - 347, where x is a year between 1943 and 2005 and y is age in years.
51x - 42 = 80; 20 7 1 x + = 4; 5 10 2 66. 21x - 52 = 70; 40 Write each word statement as an equation. Use x as the variable. Find all solutions from the set 52, 4, 6, 8, 106. See Example 5. 67. The sum of a number and 8 is 18. 68. A number minus three equals 1. 69. Sixteen minus three-fourths of a number is 13. 70. The sum of six-fifths of a number and 2 is 14. 71. One more than twice a number is 5. 72. The product of a number and 3 is 6. 73. Three times a number is equal to 8 more than twice the number. | 677.169 | 1 |
Mathematics
10A is designed for prospective elementary school teachers.� The
course covers the development of real numbers including integers,
rational and irrational numbers, computation, prime numbers and
factorizations, and problem solving strategies.� This class does
not satisfy G.E. math requirements for non Liberal Studies majors
at CSUF.� (AA, CSU)
Mathematics
15 is an intensive course covering those topics traditionally
found in the separate courses of trigonometry and college algebra.�
This course will include in-depth analysis and application of
linear, quadratic, polynomial, rational, exponential, logarithmic,
trigonometric functions and their graphs, systems of equations,
and analytic geometry.� (AA, CSU, UC)
All courses
numbered 30/60 are designed to permit department to meet an immediate
student or community need, to explore newer methods in teaching
a subject, to offer courses which are innovative, and to provide
variety and flexibility in curriculum.� A required course description
identifies each course subject.� These course
may be taken for CR/NC.
Mathematics
courses numbered 49/99 are designed for students who wish to undertake
special projects related to mathematics.� Students, under instructor
guidance and acknowledgement, may pursue individual exploration
after completing or while currently enrolled in at least one course
in the department of directed study.� (AA, CSU)
Mathematics
64 is the intensive coverage of elementary and intermediate algebra
in one semester.� This course is designed for students who have
had one year of high school algebra or equivalent and have a facility
for learning math.� This course will satisfy the intermediate
algebra requirement for any transfer level math course.� (AA)
MATH
75
ADAPTIVE
MATHEMATICS� (.5 - 1)
Class Hours:
3 Laboratory
Mathematics
75 is designed primarily for students with learning disabilities.�
It covers the fundamentals of mathematics including whole numbers
and the operations, addition, subtraction multiplication and division.
Mathematical concepts will be taught in the context of life skills
development.� May be taken as many times as needed to meet objectives.� (AA)
MATH
87
MATHEMATICS
FOR LIFE� (3)
Class Hours:
3 Lecture
Mathematics
87 consists of a quick review of common fractions, decimals and
percents; consumer applications, basic operations of algebra;
simple equations; formula manipulation; and basic facts and formulas
from geometry.� (Students who have received credit for MATH 61
will not be granted units for this course).� (AA) | 677.169 | 1 |
The Pleasures of Probability (Undergraduate Texts in Mathematics)
Delivery: 10-20 Working Days
The ideas of probability are all around us. Lotteries, casino gambling, the al most non-stop polling which seems to mold public policy more and more these are a few of the areas where principles of probability impinge in a direct way on the lives and fortunes of the general public. At a more re moved level there is modern science which uses probability and its offshoots like statistics and the theory of random processes to build mathematical descriptions of the real world. In fact, twentieth-century physics, in embrac ing quantum mechanics, has a world view that is at its core probabilistic in nature, contrary to the deterministic one of classical physics. In addition to all this muscular evidence of the importance of probability ideas it should also be said that probability can be lots of fun. It is a subject where you can start thinking about amusing, interesting, and often difficult problems with very little mathematical background. In this book, I wanted to introduce a reader with at least a fairly decent mathematical background in elementary algebra to this world of probabil ity, to the way of thinking typical of probability, and the kinds of problems to which probability can be applied. I have used examples from a wide variety of fields to motivate the discussion of concepts.
Similar Products
Specifications
Country
USA
Author
Richard Isaac
Binding
Hardcover
EAN
9780387944159
Edition
1st ed. 1995. Corr. 2nd printing 1996
IsAdultProduct
ISBN
038794415X
IsEligibleForTradeIn
1
Label
Springer
Manufacturer
Springer
NumberOfItems
1
NumberOfPages
244
PublicationDate
1996-10-30
Publisher
Springer
Studio
Spr | 677.169 | 1 |
To familiarize students with the basic concepts of MTH 211 and to make them able to develop an understanding of mathematical concepts that provide a foundation for the mathematics encountered in Engineering. The course allows students to work at their own level thereby developing confidence in mathematics and general problem solving. On successful completion of this course the student will be able to:
1.demonstrate a sound understanding of a number of mathematical topics that are essential for studies in Engineering;
2.interpret and solve a range of problems involving mathematical concepts relevant to this course ;
3.Effectively communicate the mathematical concepts and arguments contained in this course. | 677.169 | 1 |
Mathematics term paper
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Mathematics term paper
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Mathematical Methods
This course lays the foundation of all other courses of Statistics / Agricultural Statistics discipline by preparing them to understand the importance of mathematical methods in research. The students would be exposed to the basic mathematical tools of real analysis, calculus, differential equations and numerical analysis. This would prepare them to study their main courses that involve knowledge of Mathematics. | 677.169 | 1 |
Maths
The Mathematics faculty provides an academic, supportive environment that encourages students to achieve to their highest level. This is done by providing the students with course options which best suit their needs and a range of extra curricula activities in and out of the class to promote interest in the subject. Students are challenged through enrichment and extension material and engaging curriculum programs. Technology is regularly used to assist Mathematics teaching and learning.
The world of mathematics introduces students to the infinite possibilities of life. Our highly skilled staff lead our students on a journey which not only provides them with invaluable life skills but fosters the development of scientific thought. The basic components of numeracy, algebra, geometry, measurement, statistics and problem solving are the foundations of mathematical study in Years 7 and 8.
Stage 6 (Year 11) Levels offered for Preliminary study are Extension 1, Mathematics and General Mathematics. Students should choose a Mathematics course suited to their ability and interest. It is critical that students consult their Mathematics teachers in order to determine which course suits them. Extension 2 Mathematics may be available in Year 12 to students who studied Extension 1 Mathematics in the Preliminary year and continue to do so in the HSC year. (Students achieving results over 80% in Extension 1 will be invited to enter Extension 2, other students may apply.) | 677.169 | 1 |
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2007 2016 Collins Learning, a division of HarperCollins Publishers Ltd, registered in Scotland, Company No. Nno, Y. Post navigation Previous Next Everything that you wanted to know about CBSE, ICSE, IGCSE, and other international syllabiA 2 page worksheet containing problems which need to be solved using quadratic equations. Inagawa, K. pcos review article Collins Bilingual Dictionaries support language learning in secondary schools and beyond for a range of languages in a variety of dictionary styles to suit different. Genki An Integrated Course in Elementary Japanese Answer Key Second Edition (2011, E. Gistered address: Westerhill Road, Bishopsbriggs. No, C. Post navigation Previous Next Everything that you wanted to know about CBSE, ICSE, IGCSE, and other international syllabiCollins Bilingual Dictionaries support language learning in secondary schools and beyond for a range of languages in a variety of dictionary styles to suit different. Kashiki)Collins Bilingual Dictionaries support language learning in secondary schools and beyond for a range of languages in a variety of dictionary styles to suit different. Eda, Y. L can be solved using simple factorising without the need for the. To link to this poem, put the URL below into your page: Song of Myself by Walt. The General Certificate of Secondary Education (GCSE) is an academically rigorous, internationally recognised qualification (by Commonwealth countries with education.
Joy proficient essay writing and custom writing services provided by professional academic writers. Post navigation Previous Next Everything that you wanted to know about CBSE, ICSE, IGCSE, and other international syllabiA 2 page worksheet containing problems which need to be solved using quadratic equations. L can be solved using simple factorising without the need for the. Question:I'M LOOKING FOR gcse maths grade 6 past (OLD)question papers Answers:Here is a link to past maths papers for GCSE level from AQA board Maths. We provide excellent essay writing service 247. | 677.169 | 1 |
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Summary
The Nuts and Bolts of Proofs instructs students on the primary basic logic of mathematical proofs, showing how proofs of mathematical statements work. The text provides basic core techniques of how to read and write proofs through examples. The basic mechanics of proofs are provided for a methodical approach in gaining an understanding of the fundamentals to help students reach different results. A variety of fundamental proofs demonstrate the basic steps in the construction of a proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems. * New chapter on proof by contradiction * New updated proofs * A full range of accessible proofs * Symbols indicating level of difficulty help students understand whether a problem is based on calculus or linear algebra * Basic terminology list with definitions at the beginning of the text | 677.169 | 1 |
Course Summary
Entertaining lessons in this engaging course can help you study for and excel on the Pennsylvania Algebra I Keystone exam. Multiple-choice quizzes, practice exams and other resources are available to improve your algebra knowledge and enhance your preparations for the exam.
About This Course
This self-paced course serves as a fantastic resource for individuals who want to strengthen their ability to pass the Pennsylvania Algebra I Keystone exam. Fun lessons closely examine a variety of algebra topics you could see on the test, including scatter plots, polynomials, basic statistics and expressions. Short quizzes are available to assess your comprehension of those lessons, while practice exams can test your knowledge of entire chapters. Taking advantage of these study resources can improve your knowledge of topics that include:
Real numbers, rational numbers and irrational numbers
Exponential, radical and absolute value expressions
Math problem-solving and estimation
Solving problems with polynomials and systems of equations
Writing, graphing and solving linear equations
Graphing and solving linear inequalities
Understanding functions and scatter plots
Basic statistics and the probability of compound events
How to interpret and graph data
About the Exam
The Pennsylvania Algebra I Keystone exam is used to test students' algebra proficiency and serves as one component of Pennsylvania's high school graduation requirements. The test is broken into two modules that cover operations, linear equations and inequalities, linear functions and data organization. Each module consists of 18 multiple-choice and 3 constructed-response items and is worth 30 total points or 50% of the total score. The test is typically offered in December, January, May and late July.
Pennsylvania Algebra I Keystone Exam Preparation & Registration
Review any lessons in this course that can boost your knowledge of algebra topics you need to understand most. Ensure you comprehend front-end estimation, set notation, simple linear regression, quantitative data and much more. When ready, find out how much you understand about these topics by taking short lesson quizzes, practice chapter exams and a final course exam. All resources in this course are available 24/7 and can be viewed using any computer or mobile device.
To ensure students are able to take the test during the designated testing period and know what to expect on the day of the exam, be sure to contact school officials or district administrators for more information.
Pennsylvania Keystone Exams is a registered trademark of the Commonwealth of Pennsylvania | 677.169 | 1 |
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Developmental Math 4 - Advanced Algebra
Developmental Math 4 - Advanced Algebra
This course prepares learners for trigonometry and for whatever their next steps are in mathematics. It provides a more in-depth study of nonlinear equations, inequalities, and functions, while also covering induction, sequences, and counting. Other units include functions and conic sections. The units are intentionally sequential and use a spiraling curriculum approach to ensure that each unit builds on the previous units. The course is engaging, rigorous, and highly interactive, and was developed using proven pedagogical methodologies. | 677.169 | 1 |
Calculus Assessment Topics
The following list of objectives was developed during the designing of the SFU's Calculus Readiness Test. The purpose of the test is to determine whether incoming students have the necessary background to succeed in the first Calculus course they will be taking.
We placed an objective on the list below if it satisfied two criteria: 1) it was stated or implied in the B.C. high school curriculum documents; and 2) the objective is relevant for Calculus courses offered at SFU. Note that the first sixty four objectives on this list also appear as objectives for the SFU Q Placement Test. While not all of these may be tested on the Calculus Readiness Test, they represent, nevertheless, knowledge required to succeed in a Calculus course.
Some of the objectives are quite general, while others are more specific. In an effort to keep the list to a manageable number, we have attempted to refrain from being very specific except in those areas where experience suggests such specificity would be helpful.
The actual test has 30 questions, and will therefore only cover a subset of the list. However, we anticipate that the assessment test will evolve and change as time goes on, and we included items as objectives whether or not they appear on the first version of the test. Some test questions may be randomly drawn from a pool of possible questions of comparable difficulty but covering different objectives on the list.
Use fractions, decimals, and percents to solve problems;
Display skills such as simplifying a complex fraction and finding a percent equivalent to two or more sequentially applied percents;
Compare and order fractions, decimals, and percents and find their appropriate locations on a number line;
Use the correct order of operations in situations where more than one operation is performed;
Interpret exponential notation and use laws of exponents for variables with integer exponents;
Estimate the results of numerical computations and judge the reasonableness of the results;
Verify the reasonableness of numerical computations and their results in a given context using an appropriate number of significant digits;
Recognize and generalize numerical, geometrical, and other patterns;
Plot linear and non-linear data, using appropriate scales;
Given a verbal, graphical, or algebraic representation of a relationship, express it in a different form as required;
Determine whether a relationship is a function;
Distinguish among linear, exponential, and power functions;
Compare linear and non-linear functions with respect to their rates of change;
Interpret the meaning of intercept and slope of a linear function in a given context;
Interpret the meaning of the intercepts and vertex of the graph of a quadratic function in a given context;
Translate a verbal statement into algebraic language;
Translate an algebraic statement into words;
Evaluate algebraic expressions for specified values of the variables, including cases where a variable may take on negative or fractional values, and recognize that -x does not have to be negative and 1/x might be greater than 1.
Solve simple problems involving rates and derived measurements for such attributes as velocity and density;
Use graphical representations of data to solve problems;
Find, use, and interpret mean, weighted mean, or median as appropriate in the context of a given problem;
Use principles of probability to make and test conjectures about the results of experiments and simulations;
Compute probabilities for compound events;
Construct sample spaces and distributions in simple cases;
Use the concepts of conditional probability and independent events in problem solving;
Differentiate between inductive and deductive reasoning;
Interpret and correctly use connecting words, such as "and", "or", and "not";
Use examples and counterexamples to analyze conjectures;
Distinguish between "if-then" and "if and only if" statements;
Determine whether two statements are logically equivalent;
State and interpret correctly the negation of a given statement;
Analyze the validity of an argument;
Given a relationship defined by a table, graph, or formula, tell whether or not it is a function;
Given a representation of a function in any of the following forms: a table of values, a graph, an equation or formula, or a verbal description for a function;
generate a different form of representation, as required, using function notation if appropriate;
For any of the representations named in #2, distinguish between input values and output values, and/or provide an input value associated with a specified output value. Evaluate a function given in function notation for a specific value of the variable;
Use the graph of a function to tell whether or not it is increasing (decreasing) on a specified interval;
For linear functions: transform from the form ax + by = c to slope-intercept form and vice versa;
given two points, or the slope and y-intercept, find the equation;
Given two lines that are graphed on the same set of axes, tell which line has the greater (lesser) slope. Interpret the slope of a line as a rate of change. Find the point of intersection for two given lines by graphing or by solving the appropriate system of equations;
Determine from their equations whether two lines are parallel, perpendicular, or neither. Write an equation of a line parallel or perpendicular to a given line and which passes through a given point. Write the equation of a vertical or horizontal line, given one of its points;
Given a word problem or a description of a real-world event, model the situation by constructing an appropriate function or equation. Use this to find the information required, and interpret your solution in terms of the original situation. Recognize when real-world circumstances are best modeled using piecewise-defined functions;
Determine the domain and range of a function by examining the graph, the formula, or the constraints of the situation being modeled;
Given the graph of a function y = f(x), sketch the graph of y = k• f(x) or y = f (kx). Identify a graph as a horizontal or vertical stretching or compression of another graph;
Identify a graph which can be seen as a sequence of translations, or as a sequence of a translation and a stretching or compression. Given a formula for f(x), write a formula for the function after such a sequence of transformations;
Given formulas for the functions f(x) and g(x), find a formula for f(g(x)) or for g(f(x)). Given formulas or tables of values for f(x) and g(x), evaluate f(g(x)) or g(f(x)) for a specified value of x. Given the domains of f(x) and g(x), determine the domains of f(g(x)) and g(f(x));
Express a complicated function as a composite of simpler functions;
Given graphs of two functions f and g, sketch a graph of f + g or f – g. Given formulas for f and g, find the formulas for f + g and f – g;
Use the definition of |x| for any real number x to rewrite a function involving absolute values without using absolute value bars;
Determine if a given function is one-to-one, and if it is,
(a) given a formula for f(x), find a formula for f-1(x);
(b) given a graph of f(x), sketch the graph of the inverse function;
(c) evaluate f-1(b) for selected values of b from a given graph or table of values of f(x), or by using the formula;
Find the zeros of a quadratic function by factoring, completing the square or the quadratic formula;
Transform a quadratic function from standard form to vertex form by completing the square. Use the vertex form to find the maximum or minimum value of a quadratic function, or to sketch the graph without the use of calculator or computer;
Given any of the following information, find a formula for a parabola:
a) the x-intercepts and one other point;
b) the vertex and one other point;
c) the y-intercept and two other points;
Match power functions of the form f(x)=xn for n = 1,2,3,4, or 5 with their graphs, without the use of a graphing calculator or computer. Describe the behaviour of the graph of any of these functions when the independent variable is close to zero or very large positively or negatively. Use algebra to find a formula for a power function if you know two data points;
Given a formula, determine whether or not it defines a polynomial function, and if it is a polynomial, state the degree;
Use the Rational Root Theorem to find the zeros of a given polynomial function;
Given a polynomial function, identify the x- and y-intercepts. Determine the behaviour of the graph for large positive and negative values of the independent variable. Use this information to draw a rough sketch of the graph;
Given a graph of a polynomial function, find an algebraic expression for the function that might produce the graph, and justify your choices;
Given a rational function, identify any zeros or vertical asymptotes, and use the coefficients of the leading terms in the numerator and denominator to predict the behaviour of the graph for large positive and negative values of the independent variable;
Given the graph of a rational function, find a plausible algebraic expression for a function that could produce this graph, and justify your choice;
Write an exponential function to model a quantity that is growing (or decaying) by a fixed percentage in a given time period. Determine the percentage growth rate from the formula for an exponential function;
Given an expression for an exponential function, identify the domain and range, the y-intercept, and the horizontal asymptote, and sketch the graph;
Given expressions for two or more exponential functions, determine which function will have the steeper graph;
Recognize that exponential and logarithmic functions are inverses of each other. Given one such function, write the appropriate expression for the inverse. Given a logarithmic function, sketch its graph;
Relate the properties of logarithms to the corresponding properties of exponents. Apply properties of logarithms to solve logarithmic or exponential equations;
Identify an angle given in radians as a real number related to the directed rotation of a ray about its endpoint, such that if a ray with its endpoint at the center of a circle of radius r has rotated through an angle of θ radians in a counter-clockwise direction and the point of intersection of the ray with the circle has traversed an arc length of s, then θ = s/r ;
Convert between radian measure and degree measure of an angle;
Associate a point on the unit circle with a given angle θ, and define the sine and cosine of θ in terms of the coordinates of that point. Conversely, use trigonometric functions to find the coordinates of a point P on the unit circle associated with a given angle θ, or on a circle of any radius;
Sketch the graphs of y = sin x and y = cos x. Label intercepts and x-coordinates of turning points. State the domain, range, and period of the sine and cosine functions;
Determine the amplitude, period, midline, and horizontal shift of any function of the form y = A sin (t – h) + k or y = A cos (t – h) + k, and sketch the graph without using a calculator or computer;
Given a sinusoidal graph, fit a suitable function to it. Identify sinusoidal behaviour in real-world situations;
Define tan θ, cot θ, sec θ and csc θ in terms of sin θ and cos θ. Sketch the graphs of these functions, and determine the domain, range, and asymptotes for each one. Use the definitions to evaluate these functions for specific values of θ or to solve problems as needed;
Recognize and use the Pythagorean relationships among the trigonometric functions to establish identities or to solve applied problems;
Solve simple trigonometric equations and use reference angles to get all the solutions;
Given a table of values, determine the type of function which best fits the given data: linear, quadratic, exponential, sinusoidal, etc.;
Compare power, exponential, and logarithmic functions with respect to the values assumed by the function for large positive or negative values of the independent variable;
Recognize an arithmetic sequence, and identify the common difference. Calculate an arbitrary term in a given arithmetic sequence;
Recognize a geometric sequence, and identify the common ratio. Calculate an arbitrary term in a given geometric sequence;
Recognize an arithmetic or geometric series and calculate Sn if required. Use the sigma notation to write Sn. Recognize an infinite geometric series, and find its sum. | 677.169 | 1 |
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These course descriptions are not being updated as of August 1, 2016.
Current course descriptions are maintained in LionPATH.
Mathematics Education (MTHED)
MTHED 433
Function Concept in Secondary School Mathematics (3) This course develops the concept of function as an essential topic that underlies and connects school and collegiate mathematics.
MTHED 433 Function Concept in Secondary School Mathematics (3)
Prospective teachers as students need to understand the concept of function deeply as an essential topic of school and collegiate mathematics. In this course, they develop greater facility in using multiple representations and encounter function ideas as they unfold in multiple areas of mathematics, thus extending their understanding of collegiate mathematics and its connection to school mathematics. The students become conversant in current state and national expectations about functions as a mathematical entity. They plan appropriate instruction to develop secondary school student's understanding of function.
Intended as an elective for students in Secondary Education/Mathematics Education, the course helps students both to enrich and apply the pedagogical ideas and technology uses from their methods courses and to connect their collegiate mathematics experiences to school curricula. In particular, it helps to build prospective teacher's conceptual understanding of function so that they may draw more strongly on this understanding to engage secondary students in mathematics. Class activities involve use of physical manipulatives and mathematics technology (e.g., spreadsheets, geometry construction environments, and graphing calculators), as appropriate.
Students in this course would be expected to complete a major project and paper in addition to weekly assignments, exams, quizzes, and written reflections of classroom participation. Course grades depend on students' performance on all of these measures.
General Education: None
Diversity: None
Bachelor of Arts: None
Effective: Summer 2006
Prerequisite:
CMPSC 101 or equivalent; at least 18 credits of mathematics at or above the calculus level; acceptance into secondary mathematics certification program or permission of | 677.169 | 1 |
MTH 100 ALGEBRAIC CONCEPTS
This course is designed to develop and maintain proficiency in basic algebra skills and to prepare students for future mathematics work in college courses. Topics include exponents, factoring, equation solving, rational expressions, radicals, quadratic equations, graphs of functions, descriptive statistics and regression.
Credits
4 sh
Offered
Offered fall.
Notes
This course must be completed with C- or better before taking any other mathematics course. Elective credit only. | 677.169 | 1 |
I've only just started self teaching this module. The book doesn't have any mention about using a calculator, but the exam papers say you can use a graphical. So is this correct? Its just I know my friend did this whole module without a calculator and said it was like C1 in the sense the numbers workout fine and its a non-calculator paper. | 677.169 | 1 |
Normal 0 false false false MicrosoftInternetExplorer4 Finite Mathematics, Eleventh Edition is a comprehensive, yet flexible, text for students majoring in business, economics, life science, or social sciences. The authors delve into greater mathematical depth than other texts, while motivating students through relevant, up-to-date applications drawn from students' major fields of study. Every chapter includes a large quantity of exceptional exercises—a hallmark of this text—that address skills, applications, concepts, and technology. The Eleventh Edition includes updated applications, exercises, and technology coverage. In addition, modern and relevant topics such as health statistics have been added. The authors have also added more study tools, including a prerequisite skills diagnostic test and a greatly improved end-of-chapter summary, and made content improvements based on user reviews.
Arriving by train to Phoenicia, New York, in the mid-1930s, downhill skiers first discovered the snowy trails of Simpson Ski Slope. Soon after, many Borscht Belt hotels were offering skiing and skatin...
After World War II ended, control of Korea was divided between the United States, who occupied the southern part, and the Soviet Union, who occupied the north. Tensions between the two new countries e... | 677.169 | 1 |
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Description: GeoGebra is open source software that combines dynamic geometry software, computer algebra system and spreadsheet functionality. This paper explores the current literature on each of these topics i...
Geo
GeoGebra in the Secondary Mathematics Classroom 1
GeoGebra in the Secondary Mathematics Classroom 2 GeoGebra in the Secondary Mathematics Classroom Geo The paper begins with an explanation of why technology is seen as an important part of mathematics teaching and learning. This is followed by a description of the types of technologies often used in mathematics teaching and learning. Specific attention is given to dynamic geometry software, computer algebra systems and spreadsheets. Opportunities and precautions are identified for each category of software. Finally, the important characteristics of GeoGebra itself are examined. Technology in Mathematics Education – Why Bother? Technology plays an important part in the learning of mathematics. Students must become familiar with the technological tools utilized in mathematics, whether that be an abacus or a graphing calculator. Modern technology allows for easier exploration of mathematics than was previously possible. "The speed of computers and calculators enables students to produce many examples when exploring mathematical problems. This supports the observation of patterns, and the making and justification of generalizations" (British Education Communication Technology Agency, 2004, p. 1). According to the National Council of Mathematics Teachers position statement regarding technology, appropriate use of technology allows more students access to mathematical concepts (National Council of Teachers of Mathematics, 2008). A motivating factor for increasing the accessibility of mathematics is that mathematics knowledge has become as an important part of critical citizenship (Adler, Ball, Krainer, Lin, & Novotna, 2005, p. 360). To help students gain the skills that will be useful as citizens, students must
GeoGebra in the Secondary Mathematics Classroom 3 have the opportunity to use the same technology that is available outside the walls of their classrooms. (Haapasalo, 2007, p. 9). Using the same technology that is available outside the classroom allows students to transfer their knowledge into the world as they move beyond formal education. Some teachers and school systems remain wary of integrating technology into mathematics education. The three most common reasons are curriculum scope (convincing teachers the benefits are worth the change), availability of the technology (open computer labs, for example) and accessibility of the programs (technology that is easy enough to learn that the focus remains on the math) (Little, 2008, p. 49). Equipment failure can also be a major roadblock to the adoption of technology, as teachers will not commit to using something they cannot rely on in their daily teaching (Cuban, Kirkpatrick, & Peck, 2001, p. 829). The views of the mathematics teacher greatly influence whether and how technology will be incorporated into the classroom. According to a recent study, middle-aged and more experienced teachers were more likely to integrate technology than their younger counterparts, despite having a more negative attitude regarding technology (Hung & Hsu, 2007, p. 233). This suggests that familiarity with technology might not correlate to increased technology use in the classroom. A base level of technical skill is required, however, as a previous study notes that "effective teachers who use ICT [information and communications technology] are teachers who are confident with ICT" (Bramald, Miller, & Higgins, 2000, p. 5). Types of Technology Used in Mathematics Education The technology used in mathematics teaching and learning can be categorized into two major types, virtual manipulatives and general software tools (Preiner, 2008, p. 26). A virtual manipulative can be defined as "an interactive, Web-based visual representation of a
GeoGebra in the Secondary Mathematics Classroom 4 dynamic object that presents opportunities for constructing mathematical knowledge" (Moyer, Bolyard, & Spikell, 2002). Virtual manipulatives allow a student to interact with the mathematical situation without any additional skills or training required, though the student's exploration is limited by the design of the virtual manipulative. By contrast, general software tools allow the student to explore any number of mathematical concepts, but require some training to use. A variety of general software tools are used in mathematics, including dynamic geometry software, computer algebra systems and spreadsheets. Barzel defines general software tools as "tools [that] can be used for a wide set of tasks and be considered to be general purpose tools that are not useful for only a limited number of specific tasks – that is the character and as well the most important benefit of general tools" (2007, p. 81). The remainder of this literature review will be spent on examining the research on dynamic geometry software, computer algebra software and spreadsheets. These are the types of software that GeoGebra seeks to integrate into one coherent tool. Dynamic Geometry Software Dynamic Geometry Software (DGS) is the most easily adopted form of general software tools, as it was explicitly designed for classroom use (Ruthven, 2008, p. 1). DGS is controlled primarily with the mouse, allowing the basic functionality to be easily learned. Using DGS, teachers and students are able to quickly and accurately explore geometrical figures, changing their dimensions while maintaining the mathematical relationships in the figure. For example, a figure could be drawn showing a perpendicular bisector of a line segment. As the line segment is dragged, changing its position and length, the perpendicular bisector automatically moves as well. The important features of DGS are listed by KokolVoljic as:
GeoGebra in the Secondary Mathematics Classroom 5 - a dynamic modeling of the traditional paper and pencil (blackboard and chalk) teaching environment through the drag mode an option to condense a sequence of commands to form a "new command", a macro an option to visualize the paths of the movements of geometrical objects, a locus (2007, p. 56)
DGS has the ability to profoundly change the way we teach proof, one of the most crucial ideas in mathematics. DGS allows students to instantly create and test their conjectures, allowing them the freedom to explore geometry and discover patterns. Although students can easily find patterns using DGS, researchers are advising users of DGS to use exploration merely as the foundation for deductive proof, since some teachers have begun to use exploration as a replacement for proof (Hanna, 2000, p. 14). Teachers' tendency to replace formal proof with dynamic exploration is seen as a reaction to improper use of formal proof, such as only proving things students are already convinced is true (Hoyles & Jones, 1998, p. 122).
GeoGebra in the Secondary Mathematics Classroom 6 Although the power and flexibility of DGS is enticing, we must understand and acknowledge that changing the medium of teaching geometry will cause important changes in the way students construct meaning about geometry. Jones lists a number of specific areas in which DGS has a mediational impact, including: The students' understanding that the order in which objects were created leads to a hierarchy of functional dependency within a figure. The constraint of robustness of a figure under drag becoming linked with using points of intersection to try to hold the figure together. The 'dynamic' nature of the software influencing the form of explanation given by the students. (Jones, 2000, p. 80) The role of the teacher is shifted when DGS is utilized in the classroom, but the teacher's role remains critically important; the teacher's guidance is crucial as the student tries to construct meaning from the explorations they are involved in. The artefact [DGS] is exploited by a double use, with respect to which it functions as semiotic mediator. On the one hand, meanings emerge from the activity – the learner uses the artefact in actions aimed at accomplishing a certain task; on the other hand, the teacher uses the artefact to direct the development of meanings that are mathematically consistent. (Mariotti, 2000, p. 37) There are pitfalls inherent with free exploration in DGS, such as students inadvertently creating a special case by dragging a generic drawing (Sinclair, 2003, p. 291). This could lead students to incorrect assumptions about mathematical figures, such as thinking that the Pythagorean theorem holds for all triangles, when in fact it is only true with right triangles. While these pitfalls should not stop us from using DGS, we must be aware of them as we begin to incorporate the use of DGS in our classrooms.
GeoGebra in the Secondary Mathematics Classroom 7 Computer Algebra System Another type of general software being used in mathematics education is a Computer Algebra System (CAS). A CAS can be defined as "a piece of software which is capable of working symbolically as well as numerically. In principle it is a program which does on a computer the manipulation that has traditionally been done with pencil and paper" (Lawson, 1997, p. 228). CAS are primarily controlled by the keyboard through textual and numerical input. It is important to note that CAS was created for use by practicing mathematicians, not for mathematics education (Ruthven, 2008, p. 1). This has caused slower adoption of CAS into the classroom, and teachers and researchers are still attempting to come to terms with the effects of using CAS in the classroom. Much of the discussion on CAS in the classroom revolves around what portions of the curriculum students need to know how to do by hand, and what portions they can off-load to a computer. The answers to these questions greatly influence what is taught, and how it is assessed.
Figure 2. Screenshot of factoring with the computer algebra system Mathematica.
Supporters of CAS in education emphasize the ability of students to access higher level concepts, without having to drudge through tedious algebraic manipulations (Atiyah, Monaghan, & Pierce, 2004, p. 157). Access to these higher-level concepts allows students to leave contrived problems behind, giving them a chance to explore real world situations
GeoGebra in the Secondary Mathematics Classroom 8 instead (Heid & M. T. Edwards, 2001, p. 128). Leigh-Lancaster as gives a fairly comprehensive list of possible benefits resulting from the use of CAS, including the possibility for improved teaching of traditional mathematical topics opportunities for new selection and organization of mathematical topics access to important mathematical ideas that have previously been to difficult to teach effectively a vehicle for mathematical discovery long and complex calculations can be carried out by the technology, enabling students to concentrate on the conceptual aspects of mathematics the technology provides immediate feedback so that students can independently monitor and verify their ideas the need to express mathematical ideas in a form understood by the technology helps students to clarify their mathematical thinking situations and problems can be modeled in more complex and realistic ways (2003, p. 5) Despite the perceived benefits of CAS, some researchers find fault with the underlying assumption that concepts and skills can be separated. While this does not necessarily lead them to reject the notion of using CAS in the classroom, it does change the way in which CAS is used. The French researcher Lagrange is among the leading voices in this camp. For Lagrange, manual skills (or more generally, techniques) are required for the student to construct meaning. In my own experiences as a classroom teacher, I have found that if students are shown a technological solution before having a chance to practice a technique by hand, they may not ever truly understand the nature of what the technology is doing. For example, if a student is taught to multiply matrices using a graphing calculator, they may be very proficient at typing the numbers into the calculator, but may have no idea
GeoGebra in the Secondary Mathematics Classroom 9 about how to interpret the elements of the resulting matrix. I have found it to be much more effective to introduce matrix multiplication by guiding the students through a word problem and having the students define matrix multiplication themselves. In the words of Lagrange, At certain moments a technique can take the form of a skill. This is particularly the case when a certain 'routinisation' is necessary… It is certain that the availability of new instruments reduces the urgency of this routinisation… But techniques must not be considered only in their routinised form. The work of constituting techniques in response to tasks, and of theoretical elaboration on the problems posed by these techniques remains fundamental to learning. A more pragmatic concern with the use of CAS in the classroom is that students may be confused by the results given by the CAS (Artigue, 2002, p. 265). For example, when a secondary mathematics student is taught to factor a difference of cubes, they are taught a rigid algorithm, which will result in all students achieving the same answer. The CAS may or may not represent the factored form of the expression in the same manner the student is used to seeing. A student working by hand would factor as follows:
3 (8x - 27)
= (12x + 8x 2 + 18)( x - 3/2) Although these expressions are in fact equivalent, recognizing that fact may not be trivial for a student without the ability to perform such tasks mentally or by hand. This sort of situation can lead to students being unable to determine if the answers given by the CAS are reasonable (Waits & Demana, 1998).
GeoGebra in the Secondary Mathematics Classroom 10 The power of CAS will fundamentally change mathematics learning and assessment. "Whereas graphics calculators, for many teachers, slotted easily into the curriculum and enhanced their teaching with little threat, CAS demands a more thorough response" (Kendal, Stacey, & Pierce, 2005, p. 105). Paper and pencil algorithms (techniques, in the terminology of Lagrange) must be examined individually to determine if they contribute understanding for the student. Algorithms that do not contribute to a student's understanding should be performed with technology (Waits & Demana, 1998). Spreadsheets Another type of general mathematics software is the spreadsheet. A spreadsheet is simply an array of rows and columns that allow calculations to be quickly performed. More recently, "the basic paradigm of an array of rows-and-columns with automatic update and display of results has been extended with libraries of mathematical and statistical functions [and] versatile graphing and charting facilities" (Baker & Sugden, 2003, p. 19). This extension of the functionality of spreadsheets has allowed spreadsheets to become useful when teaching a variety of mathematical topics.
Figure 3. Screenshot of spreadsheet software Microsoft Excel.
Spreadsheets are useful tools for exploring a large range of mathematical topics. One of the simplest uses of the spreadsheet at a secondary level occurs when teaching statistics.
GeoGebra in the Secondary Mathematics Classroom 11 Although spreadsheets may not be suited to deal with in-depth mathematical statistics, they can be very useful for introductory level statistics, as would be seen in a secondary mathematics curriculum (Nash, 2008, p. 4603). Performing simple calculations on statistical data becomes a trivial with a spreadsheet. Spreadsheets can also be used in teaching such diverse mathematical topics as inequalities (Abramovich, 2005), limits in calculus (Abramovich & Levin, 1994) and the concept of infinity (Abramovich & Norton, 2000). Studies have shown increased student understanding of statistical graphs as a result of using spreadsheet explorations in statistics (Wu & YoongWong, 2007). As the use of spreadsheets in the classroom becomes more complex, however, new issues arise. Unless they are taught otherwise, students tend to create spreadsheets that are not reliable when cell values are changed. Students must be taught how to create spreadsheets that can solve general problems, instead of only being useful for only one specific case (Niess, 2006, p. 199). GeoGebra's Defining Features GeoGebra is software that attempts to combine DGS, CAS and spreadsheets into one application. "On the one hand, GeoGebra is a dynamic geometry system in which you work with points, vectors, segments, lines, and conic sections. On the other hand, equations and coordinates can be entered directly" (Sangwin, 2007, p. 36). Every object in GeoGebra has a representation in both the algebra window, as well as the geometry window. The user can adjust the value of the object through either representation, allowing them to either drag the geometric figure using the mouse, or change the symbolic representation using the keyboard. While GeoGebra attempts to combine aspects of DGS, CAS and spreadsheets, small annoyances reveal that this combination is done imperfectly. One such annoyance is the need to learn arcane syntax in order to show dynamic text or calculations when using the DGS feature of the software. For example, to create dynamic text that updates as the location of a
GeoGebra in the Secondary Mathematics Classroom 12 point changes, one would have to enter something like "a = " + a + "cm". By contrast, the ease of use of Geometer's Sketchpad (another popular DGS) when doing the same task allows students to show dynamic calculations without having to learn any syntax at all. Some of these annoyances may be a result of GeoGebra still being relatively new software, having been initially created by Markus Hohenwarter in 2001 as part of his Master's thesis in mathematics education (Preiner, 2008, p. 36).
Figure 4. Screenshot of finding the area of a triangle with GeoGebra.
The CAS abilities of GeoGebra are currently quite limited, though in the pre-release version of GeoGebra the CAS aspects of the software have been dramatically enhanced. In a recent post to the GeoGebra CAS mailing list, Markus Hohenwarter explained that development version (pre-release version) of GeoGebra now includes a full featured CAS system by incorporating an open source CAS into GeoGebra (M. Hohenwarter, 2008).
GeoGebra in the Secondary Mathematics Classroom 13 The development version of GeoGebra also incorporates spreadsheet functionality, though certain limitations exist. Currently, only 100 rows of data can be viewed in the spreadsheet mode. Spreadsheet cell ranges must be typed when performing calculations, not selected with the mouse. For example, one can type Mean[A1:A9], but you cannot type Mean[] and then highlight which cells you want to calculate the median of. Many advanced features of popular spreadsheet applications such as Excel are not available in GeoGebra, though most functions that would be used at a high school level are already available. GeoGebra is an open source application, which gives GeoGebra both moral and pragmatic benefits over proprietary software. GeoGebra is freely available to schools and students, eliminating the cost factor for schools with limited budgets. Students are able to use the application at home on their private computers with no site licensing concerns (M. Hohenwarter, J. Hohenwarter, Kreis, & Lavicza, 2008, p. 2). Markus Hohenwarter, the creator of GeoGebra, has stated the reason GeoGebra is released as a free, open source application is that he believes education should be free (Edwards & Jones, 2006). A side benefit of being open source is that a development community has grown around the project, which has allowed GeoGebra to be translated into many languages (39 different languages as of GeoGebra 3.0 in March 2008), making it accessible to many more students and educators. GeoGebra is written in Java, which allows it to run on virtually any platform (Mac, Windows, Linux). Being written in Java also allows GeoGebra to easily export files as dynamic webpages. This allows for simple creation of online math explorations, often called math applets. The ease with which GeoGebra sketches can be shared online has lead to a community of teachers using GeoGebra freely sharing their resources with one another online at While an online wiki of resources is useful for early adopters of GeoGebra, the majority of teachers will not begin using GeoGebra on the basis of resources being available
GeoGebra in the Secondary Mathematics Classroom 14 online. Simply providing technology to teachers does not lead to successful integration of that technology in their teaching (Cuban et al., 2001). To address this concern, an International GeoGebra Institute has been created, with the purpose of providing structured training and support to teachers interested in using GeoGebra (M. Hohenwarter & Lavicza, 2007). Various chapters have been set up across the world, allowing those interested in learning GeoGebra to have local support. Conclusion Incorporating technology into mathematics teaching and learning allows greater access to mathematical concepts. General mathematics software allows students to explore any number of mathematical situations, but require students to learn the software first. Dynamic Geometry Software is quite easy to use, allowing students and teachers to test conjectures by exploring geometrical figures. The manner in which proof is taught in mathematics has been greatly affected by the introduction of DGS to the classroom. Computer Algebra Systems are able to perform much of the symbolic manipulation that students do by hand. Educators must determine which algorithms can be delegated to a CAS and which must be done by hand. Spreadsheets are particularly useful when teaching statistics, but can also be used to teach a wider variety of mathematical topics. GeoGebra combines DGS and CAS into one application (development/future versions also include a spreadsheet). GeoGebra is open source software, which allows anyone to download the software and use it for free. As a Java application, it can run on any platform. To help teachers learn how to incorporate GeoGebra into their classrooms, an International GeoGebra Institute has been created to provide structured training and support. | 677.169 | 1 |
Category Archives: Mathematics Done in English
I first wrote this textbook in 2015 for a tenth grade English class in Japan. My students are planning to study abroad in eleventh grade. A good way to learn to do math in English is to do math in … Continue reading → | 677.169 | 1 |
Calculators: In this section, you will find links to calculators available online and tips for buying a hand-held calculator.
Diversions: Has some math-related games and puzzles. You should check back here from time to time because the content will change as the year progresses.
Glossary: A list of the mathematics vocabulary used in this course with the definitions.
NM Standards: This is the full version of the New Mexico Standards for Mathematics for Algebra. These are the concepts that you will learn in Algebra 1, Algebra 2, and advanced math courses. As a result, you will not study all of them this year.
Credits: This is a list of the sources for the images and videos that I did not make myself. | 677.169 | 1 |
Best-selling author D.M. Etter introduces readers to general problem-solving and design techniques through a five step process which uses MATLAB, the popular engineering software, for analysis and graphical display. The book features chapters organized around specific engineering applications drawn from a variety of engineering disciplines. This book is the cornerstone engineering tutorial in the MATLAB Curriculum Series | 677.169 | 1 |
F&n o level coursework sample students of the O levelCoursework in Mathematics A discussionpaper mathematics courseworkO level f n coursework, level. What is the difference between cambridge igcse and o levels. Cambridge igcse and cambridge o level are equivalent qualifications grade for A coursework option is. Hullos. Im Currently stuck on my F&N Coursework A. Teacher didn't offer any help , So here i am asking. How do i make a Overall time plan for F&N coursework A. | 677.169 | 1 |
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