Split (1)

train · ~1.24B rows (showing the first 1.67M)

text stringlengths 8 1.01M
Introduction to Statistics for AP Teacher There are summer courses that are especially suited to AP statistics teachers who need to tune up their statistical skills. They are taught by Dave Bock, co-author of Stats: Modeling the World, a leading AP statistics text. The course text is Bock, Velleman and De Veaux, Stats: Modeling the World, with ActivStats CD. You can purchase it directly from the publisher here. This text is among the top choices for AP Statistics. (Note: If you are a statistics instructor contemplating teaching with this book, you may wish to obtain the Instructor's edition, which you can find using the above Addison Wesley link). ActivStats is an award-winning multimedia statistics tutorial. It runs on PCs and Macs. Important: To make sure you get the right textbook bundled with the right CD, we strongly recommend that you buy them as a package directly from the publisher at the above link, rather than from Amazon or a used book market. Please order your text in time to have it in hand by the course starting date! Those contemplating teaching AP statistics (or any high school statistics course) will probably want to use a graphing calculator. The textbook integrates instruction in using the TI-83 family of graphing calculators and provides brief tips on using SPSS, JMP, Minitab and the TI-89 graphing calculator. Depending on prior experience with statistics, those planning to teach AP Statistics may feel a need to put in more hours than the 15 estimated in the course description, in order to be well prepared to teach this material. What our students say:Good value for the money. Thank you very much for a thought- provoking course" J. Politch, Harvard "I look forward to taking another course on statistics.com - a great way to continue learning in a structured manner, but flexible enough to participate while Life continues." B. Berg, AMPS Intl. "Considering all of the material that needed to be covered, I thought the course was well written and thought provoking."
Beecher, Penna, and Bittinger's Algebra and Trigonometry is known for enabling students to "see the math" through its focus on visualization and early introduction to functions. With the Fourth Edition, the authors continue to innovate by incorporating more ongoing review to help students develop their understanding and study effectively. Mid-chapter Review exercise sets have been added to give students practice in synthesizing the concepts, and new Study Summaries provide built-in tools to help them prepare for tests. The MyMathLab course (access kit required) has been expanded so that the online content is even more integrated with the text's approach, with the addition of Vocabulary, Synthesis, and Mid-chapter Review exercises from the text as well as example-based videos created by the authors
ALEX Lesson Plans Title: Land Surveying Project-Enhancing mathematics in the career/technical classroom and providing relevance in the mathematics classroom. Description: This Standard(s): [STM] STM2 (9-12) 9: Apply correct drafting techniques using computer-aided design (CAD) programs to produce plans, diagrams, and working drawings for the construction of models, prototypes, and final products. [MA2013] GEO (9-12) 21: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [G-SRT8] [MA2013] GEO (9-12) 23: (+) 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). [G-SRT11] Subject: Mathematics (9 - 12), or Science, Technology, Engineering, and Mathematics (9 - 12) Title: Land Surveying Project-Enhancing mathematics in the career/technical classroom and providing relevance in the mathematics classroom. Description: This
Think Through Math - Idaho Math Initiative Resource Think Through Math (TTM) is an online, supplemental math instruction and tutoring program that will help raise student achievement in Idaho by providing students with focused instruction, rigorous math problems, access to live certified teachers and a motivation program with rewards for working on math problems. Think Through Math works with the classroom teacher to tailor the online math instruction to the needs of the individual student. Students can access the program anywhere they get Internet access - whether at home, at school or in a library. Currently, schools in more than 25 states across the country are using Think Through Math as a supplement to instruction and noticing increased student achievement. Please visit to learn more about Think Through Math. The Think Through Math (TTM) computer-assisted program is now available to utilize as curriculum for Algebra 1 credit recovery and district approved Alternate Route for Graduation. The Think Through Math curriculum option can be used to facilitate the instruction of the required core math content, by an Idaho certified, highly qualified teacher of record. The programs are simple to implement and monitor, and students will go through a rigorous set of coursework. The credit recovery pathway will give students a solid foundation in Algebra 1 and will prepare them for success in Geometry and future math courses. The Alternate Route for Graduation pathway will give students a solid foundation in math skills aligned to the 10th grade ISAT blueprint. TTM's Algebra 1 credit recovery program initially enrolls students into a pathway of 62 "Target Lessons". The Alternate Route to Graduation program enrolls student into a pathway of 69 "Target Lessons". The pathways were constructed in conjunction with the Idaho SDE. All lessons for the credit recover program are aligned to the Common Core Standards Algebra 1 course and the Idaho Algebra 1 standards. The Alternate Route for Graduation lessons are aligned to the 10th grade ISAT blueprint. When the Smarter Balanced Assessment blueprint has been finalized, TTM will begin working on aligning the pathway to the new assessment blueprint; to be available when Idaho moves to the new assessment. For both programs, students begin by taking a brief placement test which will determine a student's readiness. Based on these results, additional pre-cursor lessons will be added into each student's pathway. Additionally, TTM monitors student progress and adds additional pre-cursors as needed to fill students' learning gaps. When students are working outside of the classroom, they can get assistance from TTM's live certified teachers. As with all coursework, the teacher is a key component in a student's success. The benefit of Think Through Math is that multiple students can be working toward recovering credit on a very individualized basis. An appropriately certified and qualified teacher, however, is necessary to ensure that students have the support to grasp the concepts being presented. As students work through their pathway, reports on student progress are available through the teacher portal. These easy-to-use reports show how much progress students are making toward completing the Target and pre-cursor lessons on their pathway. The Think Through Math program does not qualify as a stand-alone course such as those offered through IDLA. Any computer-based program must be supervised by a properly certified and qualified "teacher of record" who must actually be present in the classroom throughout the sessions. If Think Through Math is used for math credit recovery or alternate route for graduation, a math assignment code should be used when reporting in ISEE and the teacher of record facilitating the program must hold a math endorsement and be highly qualified in order to grant graduation credit. Please see documents below for in-depth information on both programs, including tools for monitoring student progress and grading. Also attached are documents on standard alignment. For Idaho Education Laws and Rules on High School Credit Requirements for Mathematics, please see Idaho Administrative Code 08.03.02.105.01.d. This code can be accessed online at Please be aware that a credit recovery course must meet the appropriate content standards of the original course. For Idaho Education Laws and Rules on Alternate Route for Graduation, please see Idaho Administrative Code 08.03.02.105.06, at Please be aware that alternate routes to graduation are locally established and must go before the local school board for approval. Once approved at the local level, the alternate route for graduation must be sent to the Idaho Board of Education for review. A copy of each districts' alternate plan is kept on file at the Office of the State Board of Education. For more information on the steps to establish a local alternate route for graduation, please contact Allison McClintick with the Idaho Board of Education, allison.mcclintick@osbe.idaho.gov. To utilize Think Through Math in either capacity communicated in this email, please contact Marisa Alan, TTM Regional Program Manager, at malan@thinkthroughmath.com. Students of Sacajawea Junior High School in Lewiston won a nationwide math contest sponsored by Think Through Math, an award-winning provider of online supplemental math instruction. The Idaho State Department of Education provides Think Through Math to all public school students through the Idaho Math Initiative. On Wednesday, May 15, representatives from Think Through Math will visit Sacajawea Junior High for a special event to award the students with the Think Through March "MATH" Madness trophy, banner and awards. One hundred-ninety five students from 7th, 8th and 9th grade math classes worked on Think Through Math in class and at home to earn the highest student average of completed lessons and total problems solved, beating out 27,565 schools from across the nation. Principal of Sacajawea Junior High, Mr. Uhlorn said, "We are proud of the work the staff and students at Sacajawea Junior High did during this competition. Many of our 195 team members worked daily to improve their math skills using TTM. We have used this program for the past three years as a great way to have students fill in gaps of mathematical concepts with the flexibility of individual pathways. Being part of the competition was an added bonus. The kids were so excited to hear how we progressed throughout the competition and made it their goal to keep working hard". During the contest, Sacajawea Junior High students spent 1,121 hours on Think Through Math completing 72,650 complex math problems. The junior high students completed 6,287 math lessons during the qualification and elimination rounds and logged 84 hours outside of the traditional school day. Think Through Math - New Enhancements for SY 2013-14 Think Through Math has been hard at work this summer making improvements to the program to ensure the absolute best experience for your students. These enhancements include: Redesigned student and teacher dashboards that put critical information right at your fingertips K-2 precursor lessons to provide full remediation for 3rd graders The addition of new lessons and fill in the blank answers to provide rigor and variety Auto-logout feature to enhance time on system reporting New and updated reports for teachers to ensure you get the information you need to guide math instruction Updated learning coaches and enhanced live teaching tool to ensure your students get the help they need when they need it A messaging system where teachers can send a star or a personal note to each of their TTM students TRAINING TTM training and ongoing support is available free of charge and can be provided via webinar or onsite at the district or school level. We highly recommend that all teachers that plan to use TTM attend one of our trainings- Utilizing TTM Effectively to Support the Idaho Core Standards. Please contact your regional program manager to set up district training opportunities. Informational Fliers TTM Technical Check To determine if your computer is compatible with TTM's technical specifications, sign into your teacher account and click on the username in the upper right hand corner. You will see a drop down menu where "Tech Check" is an option. Click on "Tech Check" and an automatic check will run. For full TTM technical requirements, please refer to the PDF document below. If you are having further difficulties, please contact your school or district IT support staff.
Algebra TOPIC DESCRIPTION: Algebra is the branch of mathematics concerned with using known information -- expressed as mathematical equations or similar relations -- and using it to generate new relations or solve for values of unknowns. The umbrella term "algebra" includes more elementary concepts such as the use of variables, function graphing, and graphing as well as set theory and more advanced "abstract algebraic" concepts.
Book DescriptionThis is a great book to learn linear algebra from. It introduce you for the basic of linear algebra, like matrices and how to work with them and what you can use them for. There are many matematics sentences in this book with proofs. Also you will learn how to manage determinants and eigenvectors, not forgetting the more difficult Normal Jordan Form (if you need that). The exercises are great to make you remember all the sentences. So if you want to learn more about vectors in dimensions, you can not imagine, such like 5-d, then here is your tool. Also the text is written so you can understand it, wihout beeing someone needing beeing perfect to English. Doing Maths for my Computer Science Degree wasn't really what I envisaged. So I bought this book to help me with my maths. Incredibly I didn't struggle with it at all. The theory is well explained and the examples make it easy to quickly use the newly learned in practical applications. The authors go into enough detail so you can understand the stuff, not just learn it. On the other hand, they don't warp your brain with unnecessary junk. The proofs were especially helpfull. All in all an excellent book about linear algebra today. I would recommend this book to anyone who wants to gain a good understanding of the basics of Linear Algebra. It has many examples to accompany the theory which help the student prepare for exams and also the added bonus of MATLAB-a computer program that comes with the book that allows the user to perform many of the calculations gieven in the book. Also answers given at the back. 5.0 out of 5 starsOne of the greatest introductory books in Linear Algebra2 Feb 2009 By Arash Mehraban - Published on Amazon.com Format:Paperback This book is very well written along with many great examples. This book covers basically everything you need to know in an introductory Linear Algebra class about Vectors, Vector spaces, Matrices, Determinants, Linear transformations, Eigenvalues and Eigenvectors, orthogonality, Gram Schmitt process, Systems of equations and and so forth. The people who think this book is too hard, should go back and review their high school algebra books again. One of the great features of this book, as an INTRODUCTORY book, is that it teaches you everything in R^n space. Overall: great 5 star book. 5 of 6 people found the following review helpful 1.0 out of 5 starsSolution Manual incomplete.1 Sep 2005 By S. Richardson - Published on Amazon.com Format:Paperback The book itself is difficult to read. When new symbols are first used, the explination are burried in the text and time consuming to find. The use of dark blue ink to "color" the items they are refering to is very difficult to distiguish from the normal black text. I waste more time trying to figure out what the symbols mean than I spend understanding the concepts. The solutions manual covers every 3rd problem, not the compleete set of odd problems like it should (eg. 1,4,7,10...). Detail on the problems is average at best. 7 of 9 people found the following review helpful 4.0 out of 5 starsNot the best for beginners, but still good2 Oct 2002 By A Customer - Published on Amazon.com Format:Paperback After giving up on this book when I first learned Linear Algebra, it has turned out to be a very good reference book on the subject. If you need to, use Anton's book to learn the subject, but I keep this one on my shelf. 4 of 5 people found the following review helpful 3.0 out of 5 starsA Fair Treatment5 Feb 2009 By math112 - Published on Amazon.com Format:Paperback The book presents linear algebra problems pretty much in outline form---an "ok" treatment for some, but not all, of the subject matter. Nevertheless, there are still many topic areas where the explanations are not thorough enough. Also,its treatment of theorems is not accompanied by a step-by-step explanaton of a problem that relates to it. After all, this book is supposed to be an aid for students taking this subject for the first time. I have found a few editing type errors--in some problems: wrong signs, numbers misplaced. This seems to be an omnipresent problem with many of these type books. WHO IS OUT THERE DOING THE FINAL REVIEWS? I think this subject matter needs some type of video teaching component, but good luck trying to find one. I think the book has its advantages, but mainly to those already immersed in the subject matter of matrices and determinants, vectors and space. A didactic tool it is not! 1 of 1 people found the following review helpful 1.0 out of 5 starsNot a good book.10 Nov 2012 By Bill - Published on Amazon.com Format:Paperback When I first looked at this book I was purchasing it for a class. I saw that about half of the people rated as 5 stars which is about 50%"ish" of the total ratings. It seems like a lot of books on the topic of LA get tough reviews so I didn't consider purchasing supplemental study material like I do for most other classes. After studying this book and figuring out how much I hate it, I started trying and figure out why so many people like this book. I discovered that 3 people were actually rating the seller, not the text itself. So here is what I don't like about this book. There is a proof is section 3.2 for a vector subspace of polynomials. Using their same logic for their proof on showing the defined problem was not a subspace, you could also disprove everything they claimed to be a subspace. The author also defines problem using fictional methods of addition and multiplication for the sake of a proof which doesn't help me understand the topic. It's hard to explain, but there is a lot of irrelevant examples. This isn't actually in the book but I feel like a lot of the examples are posed in the context if "If 2 + 2 = 5" .... Well 2+2 is never = to 5 so why are we even discussing it to show how something fails under given properties. I feel like the proofs fail during the initial statement therefore it was point to begin with. There are several examples where you see stuff like A = s[a+b], T(A) = [sa+b]. Again, this is not possible so why are we discussing this as a proof? I just can't find the logic in this book. If you have a choice, I would recommend Linear-Algebra-Its-Applications-Edition by David Lay. It still has a lot of the mathematic rigor. The book was also written with in the last decade so it familiar when you are reading it. The big thing that I like he shows you how linear algebra is applicable. I can now balance a chemical equation using Gauss-Jordan row reduction which makes total sense.
Algebra seems to be the minimum popular between all the branches of math. The combination of letters, values, and numbers on both sides of the equal to sign puts students into a lot of misunderstanding and confusion. Beginning with pre-algebra and finish with algebra 2, high school algebra covers all the basics and fundamental concepts and goes into some more detail to prepare students for their collage level algebra or math courses. Online Algebra tutor is a very good option for students who are facing difficulties in keeping up with algebra. More or less students do not have a good mathematical mind and need extra more time to understand any difficult algebra or mathematical problem. Hence, they also feel very much hesitation to ask questions in a classroom environment but Online Algebra tutor helps and gives them a convenient and easy environment to solve their problems. Also they can take many sessions with the same tutor. Some particular students need personalized attention for Math and especially for Algebra. So the Online algebra tutor is the most convenient and smart choice for students. Students can comfortably and very easily share their Algebra problems and difficulties with tutors by using online tutoring which is very conveniently and easily accessible using a computer or laptop and an internet or broad band connection. Online algebra tutor help covers all essential and important topics like Division of Polynomials, Radicals, Polynomials, Factoring, systems of Equations, complex numbers, and others of different grades. Online algebra tutor helps teach satisfactory knowledge on each topic and hence, students can get a great score in exams. Also free online Algebra worksheets and calculators make learning and understanding the subject simpler and easier and also give students so much practice to get superior in Algebra and other topics. There are massive amount of software available on internet absolutely free of cost and works wonders for your computer. Here is top 10 software list which will make computer a better experience without costing anything to you. 1. Firefox- is the most famous and greatest internet browser on internet. Easy, simple but packed with awesome features and superb speed. It lets you relish your browsing experience all the time. 2. AVG Antivirus- is very popular antivirus software, which is mostly used for the protection of computer by the users around the globe. Although it is not only software available on internet for virus protection there are so many more name which comes into mind when we talk about antiviruses like McAfee and Norton which are also very famous and popular antivirus software's. 3. PC Tools Firewall- it is a great program which is guaranteed not to give you a headache. And very user-friendly software, very easy to understand and absolutely free of cost. In just a short time guards and protect your system from all external attacks. 4. Malwarebytes- it is very modern looking software which detects and removes all malware threats in the computer. It monitors every process in your computer cautiously and protects from the malicious process. It is developed on the latest technology trend which allows this great software for fast detecting, destroying all the spywares and malwares. 5. CCleaner- it has become very much used software in recent years. It is also freeware software of windows which enables the user to maintain good control of its computer. 6. 7-zip- a file archiver with a great compression ratio and can unpack almost every format like GZ, TAR, and ZIP etc. 7. Ultimate Windows Tweaker- it is absolutely free to download, used for tweaking and optimizing windows vista and windows 7. 8. IrfanView- offers so many tools to edit both multiple media files and images with ease. 9. Notepad++- it permits user to manage easily more than one document simultaneously. 10. Picasa- it is superb free photo organizing and editing software from Google.Because of its simple interface and great features it is prefect for both beginners and advance users. T-shirts have remained and are nonetheless likely the most favored garments for just about every sexes; this really is as a result of their comfort degree plus the ease with which just one can use these inside their day-to-day lives. The recognition of T-shirts continues to be producing with all of the passage of time and along with the advent of new design trends. The other concern of their reputation may be the adoptability, they are made for just about each personal ranging from smaller infants and small ones to grownups and elders, they can be found in lots of colours and designs. Once we speak concerning the T-shirts, we cannot ignore the recognition of Blank t shirts. When sporting T-shirts is so swift, then purchasing for just about any blank t-shirt or wholesale t-shirts is even significantly less complicated. You'll discover an abundance of internet websites over the internet which have a broad selection of black t-shirts made available at pretty affordable costs. In conjunction with offering the blank t-shirts, these on the internet retailers also have a broad assortment of organic baby clothes, as well as merino wool clothing and merino wool thermals. Organic baby clothes can also be gaining recognition in current occasions as a result of reality they may be hypoallergenic in nature and specially for infants and little ones that endure from asthma, some allergic reactions or have a delicate or delicate skin. Overall health and fitness conscious dad and mom usually appear out for baby or young children clothes that are produced up of organic cotton. You are going to locate an abundance of on the net clothing stores that provide this kind of kind of baby clothes and at some discounted charges. Together with the climate taking turns, sometimes it truly is scorching and every so often it is cold, you are going to need to have your clothing ready to suit the climate predicaments. Below severe cold conditions you are going to require further protection to help continue to keep the body temperature warm. The merino wool clothing is specially developed to provide you that further safety in winter season. One of the most famous variety of merino wool clothing could be a merino wool thermal acknowledged greatest to deal with that additional pinch of cold. Film-maker and Director Franck Gigante is actually a Hudson county indigenous, who was born in Hoboken inside the yr 1968. Simply since he was 7 he dreamt of getting to be an actor. His moms and dads and close friends discovered the opportunity to complete a single matter towards the movie industry in just one way or an extra. His intentions of getting to be an actor are area on keep, as staying the oldest member among his brothers and sisters he had to work in his early age. But while operating, he by no implies left his dream of starting to become an actor die. He retained on pursuing his dream in only one way or a distinct. Due to the fact the several years passed by, his passion of getting to be an actor received additional intense and in the twelve months 2005, he lastly been given his palms on satisfying his lifelong desire. He obtained the option to get started on his acting occupation through the Puglia Area of Italy. He was introduced a terrific deal of roles within just a collection of comedian and remarkable fictions; he also obtained roles in certain tunes movies for Irene Grandi "Non-resisto" in addition to a Pluriel Swift which was presented within the fifty sixth Venice movie festival. . Some functionality movies like L 'ariamara and Frankie Facciatosta & the Search for Accomplishment also witnessed his acting capabilities. Later he was presented an risk to show his skills in many Puglisi fictions far too as sitcoms, also as in assortment shows'. In 2007, Frank started to position on trial his abilities as director inside the quick motion picture named "Role Play". Loads of appreciation and self-satisfaction from his first experience Franck developed interest in the field of direction of other Shorts, like "Mamma's Heart" and some others. Recently, Frank Gigante Filmaker has moved back to New Jersey to additional his profession not only as an actor but also as getting a motion picture director and film maker. In 1979 Richard Beyer a renowned sculptor developed most favored interactive artwork for Seattle. It recollects and present respect forInterurban the light rail line that was employed to attach the neighborhoods with Seattle. The art piece includes a curious human confront puppy and five folks standing beneath a shelter. It really is madeof forged aluminum; the sculpture has paying out focus the mischief, creativity and creative imagination of a large variety of neighbors each and every 12 months. The Interurban has hosted displays, costumes and "art attacks" celebrating all from birthdays, weddings, congratulations, bon voyages, memorials, I-love-you, friendships and excellent instances to demonstrations and well-known triggers. This nonstop stream of interactive mastermind has created it the Fremont's most identifiable landmark as well as a indicator in the entertaining and imaginative vitality for which the world is becoming renowned.Now any person may get in that Fremont's entertaining system. The one rule listed here is every person cleans up when their screen is finished and no business messages. The identify of sculptor is Richard Beyer whose operates are well renowned and broadly exhibited during Fremont. Beyer's intention was to produce this a general public artwork that might be used, appreciated and photographed for almost any goal; Richard is from rural Pateros, in jap Washington. The explanation driving human confront puppy is yet another legend, Arman Napoleon Stepanian. You'll find some suggestions that has for being accompanied by each and every citizen to take part in Decorating the Interurban. 1. BE well-mannered 2. First appear and you also can very first to embellish. No room reservation for loved ones or pals. three. No publicity slogans or logs or terms must be for being identified within the statue 4. Without certain authorization in the artist, no-one can do promoting five. Ribbons could be remaining for around complete week even though large rains will injury decorations and when you achieve your vacation spot to discover the ribbons over and above injury you could get absent them to position your own personal. 6. Please remember - BE Polite - the sculpture is planned to the satisfaction of all. Hair extensions are parts of hair which can be possibly hooked up in your scalp or along with your organic hair as a way to make your hair search for a longer time and delightful. The craze of acquiring different kinds of hair extensions is attaining reputation day by day. You'll be able to very easily locate distinct types of hair extensions available in the market today. Human hair may be the most often employed as well as the costliest sort of hair extensions employed since it seems essentially the most organic. The organic human hair extensions blend effectively along with your organic hair. You'll be able to very easily clean and shade these extensions as your organic hair. Now you'll be able to take pleasure in possessing true curly Brazilian hair with your head through the use of Brazilian hair extensions, with all the new bonding strategies and vast availability of cheap Brazilian hair which is by natural means curly without having any want for processing. A single point to maintain in mind is if you choose to go together with the Brazilian virgin hair get them hooked up by a effectively skilled skilled who must observe each of the safety measures just before attaching them in your skin. Yet another edge of acquiring them hooked up from a effectively skilled skilled is they will likely be capable of make them indistinguishable from the very own hair. When you've got your Brazilian hair extensions hooked up professionally there's no want to worry, you'll be able to take pleasure in your typical existence regimen. You'll be able to take pleasure in swimming, you'll be able to clean them with shampoo along with your typical shampoo and also you'll be able to comb them like your own personal organic hair. Should you get excellent treatment of the hair extensions, you'll be able to take pleasure in donning them for for a longer time period of time of your time could be six months into a 12 months. Each and every from time to time you ought to implement conditioning and remedy in your hair extensions as a way to boost their existence. Is there a cure for herpes? Will there ever be a cure for herpes? Is there ever going to be a cure for herpes? All of these concerns are requested every so often from the individuals that suffer from this illness. The very first thing to understand is use out some details concerning this illness like what's it, what exactly are some of its signs and symptoms, how 1 can cope with it and clearly what exactly are numerous answer or remedies to remedy or get rid of herpes. What's Herpes? Herpes is really a bunch of viral bacterial infections brought on by a particular pressure of virus known as the herpes virus. Amongst numerous STDs that are typical in people, herpes is 1 of probably the most harmful 1. Herpes is really a complicated illness but when couple of actions are taken 1 can chorus from obtaining it. Anybody struggling with this terrible illness understands how unpleasant and deliberating an outbreak may be. Creating issues even worse, most of the remedies which are accessible these days don't completely deal with the illness. Therapy Individuals inquiring the query is there a cure for herpes ought to bear in mind the most significant factor to know would be that the herpes virus isn't regarded as to become lifestyle threatening, it could nonetheless place with each other extremely aching, extremely irritating signs and symptoms. Another most significant factor to bear in mind for individuals inquiring will there ever be a cure for herpes would be that the herpes virus has the possible and capability to obtain transmitted if the contaminated individual goes with the herpes outbreak or even the virus continues to be dormant. There isn't any remedy for herpes, but there are methods and techniques that assist to suppress the herpes virus to ensure that these impacted can stop the recurrence of typical signs and symptoms related with herpes this kind of as sores and warts. It truly is human nature; men and women are likely to want that far more which they can not have. There are many specialist corporations like Banking companies, Educational institutions, and Universities and many others which tend not to let the liberty of exciting on the personnel for the duration of operate. The feel that downloading of seeing porn or actively playing game titles are not able to be permitted underneath any situation while in the business office environment. Often it applies to nations around the world at the same time, some nations around the world will block specified sites sort their citizens. The aim on the proxy internet sites is usually to bypass these filters. These proxy internet sites also support folks who tend not to want their searching practices to get witnessed or viewed. You can find a number of people who will be hunting to own the best proxy site. The www entire world incorporates a wonderful offer of the form of sites, but in relation to buying a proxy internet site that is certainly have faith in deserving and safe can be a complicated process. The reason being that some free proxy site are not able to be dependable even though there are various other folks which may not be of your respective liking. You may need to look at them carefully, ahead of finalizing which one particular to pick. One among the most important factors to search for while in the best proxy site would be the spot from which the proxy is hosted. proxking.com The top area to search for a free proxy site is Google. It provides countless internet site addresses once you begin seeking a proxy internet site. After it has yielded the search results, another action is usually to figure out in regards to the status on the internet site you have in your mind. List of internet site As proxy internet sites give usage of internet sites that are supposed to be blocked, so there will constantly be described as a possibility which the those who blocked the initial internet site to start with may even try and block your proxy internet site at the same time eventually in time. There are many kinds of hackers which are well aware of various techniques to hack facebook account or hack facebook password. By utilizing the various kinds of hacking tools, the hackers can quickly hack anyone's facebook account within minutes. In case you would like to get information about hacking there are so many websites on the net that can give you such information. One of the easiest ways to hack a facebook account is through the use of key loggers; following are some of the tips that show how to hack facebook account through a key logger. Key logger is basically a software program, which can be found on any popular software download website. The function of the key logger is that it resides in your computer device and keeps track of all the key strokes so when you enter a password it records them as well and eventually makes a report which could be seen and retrieved by a hacker without you even knowing it. So yes, key logger can be considered as a good tool for hacking a facebook account. There are also hardware key logger available in the market to serve the purpose but they are bit expensive than the software. The hardware key logger resembles a flash drive, which contain key log software in it. All you have to do is attach the device t the victim's computer and it will start extracting the data akin to the USB flash drive. Hacking a facebook account through key logging though seems easy but it pertains to enormous risk as well because if anyone sees you doing that you can pray for not being whip-lashed by the other person. Phishing is another technique that is commonly used for hacking facebook accounts at the beginner's level. Portable water heater is applied in a assortment of popular day-to-day home and industrial wishes, from washing dishes to cleaning floors; scorching water includes a multitude of utilizes. Because the very good top quality of supplies and insulation values are commonly the identical for portable water heaters, selection is purely dependent about the demand for which the sizzling water is wanted and area available to the unit. There are many problems that we can advantage during the portable water heater by way of illustration it gives comfort from cold water, it can be feasible to take a good cozy bath with sizzling water, give us hot coffee or tea and can be made use of for a range of numerous home functions. Selecting out between propane vs. electric portable water heater involves currently being mindful of the positive aspects and disadvantages of how these two kinds of water heating procedures. The utilization of a portable water heater will not build any damaging byproducts whereas the propane water heater does. Propane water heater develop carbon monoxide which could be pretty unsafe the all round overall health. The portable water heater is much less expensive as when compared with a propane water heater. Utilizing a portable water heater is normally hassle-free as when compared with the propane water heater. The propane water heater calls for loads of servicing and one needs to confirm just about every single now and then if it could be operating effectively. The propane water heaters also carry the danger of explosion if miscarried whereas the portable water heater isn't going to. Simply because the portable water heater is utilized all yr round, consequently appropriate care need to be observed. You will find some portable water heater tips which once followed could make it final lengthy. Some water heater tips are as follows: Frequently flip off the heating thirty minutes prior to working on the portable water heater. Do a standard inspection of the many valves When checking the valves and joints look for leaks. And when you uncover some seek to fill the leaks as quickly as you can.
Product Synopsis This textbook makes use of the popular computer program MATLAB as the major computer tool to study mechanics of composite materials. It is written specifically for students in engineering and materials science, examining step-by-step solutions of composite material mechanics problems using MATLAB. Each of the 12 chapters is well structured and includes a summary of the basic equations, MATLAB functions used in the chapter, solved examples and problems for students to solve. The main emphasis of Mechanics of Composite Materials with MATLAB is on learning the composite material mechanics computations and on understanding the underlying concepts. The solutions to most of the given problems appear in an appendix at the end of the
Mathematics For MBA Entrance Examinations (Fully Solved) 22 Edition is a step-by-step book for aspirants of MBA entrance examinations. It consists of questions with solutions and short-cut solving techniques. Summary Of The Book Mathematics For MBA Entrance Examinations (Fully Solved) 22 Edition is a guide for those who are preparing for their MBA entrance tests. It is a collection of a large number of objective type questions with their solutions. The book consists of short-cut methods that can be used by candidates to solve questions with much more ease. This book is of huge help to MBA aspirants. Mathematics For MBA Entrance Examinations (Fully Solved) 22 Edition is divided into 7 sections, each consisting of a set of chapters each. The first section is Algebra and the chapters that are covered in it are Basic Operations and Factorization, Linear Equations, Quadratic Equations, and Set Theory. The second section of the book covers Arithmetic. Some chapters that fall under it are Decimal Fractions, Square Roots, Percentage, Profit and Loss, and Unitary Method. The third part of the book covers Menstruation, Progression, and Probability. The fourth section is on Trigonometry and has two chapters, namely T-Raions and Identities, and Heights and Distances. The fifth portion of the book concentrates on Geometry, covering topics such as Polygons, Circles, Triangles, and Loci. The next section is about Co-Ordinate Geometry and the last covers Statistics. About R.S. Agarwal R.S. Agarwal, born in 1946, is a popular author. The author has written more than seventy-five academic books. He writes for Nursery classes to graduate degree subjects. His books are also aimed at students preparing for competitive exams. Some of his books are Quantitative Aptitude For Competitive Examinations 19th Edition, A Modern Approach To Verbal & Non-Verbal Reasoning Revised Edition, A Modern Approach To Logical Reasoning 1st Edition, An Advanced Approach To Data Interpretation 11th Edition, and Advance Objective General Knowledge 1st Edition. He completed his graduation from Kirorimal College in Delhi and after his master's joined at a lecturer's position at the N.A.S. College in Meerut. After winning a fellowship for three years in 1976, he joined the Delhi University to do his Ph.D. After that he got promoted at the NAS College. In 2003, Agarwal took voluntary retirement. The 'certified buyer' badge indicates that this user has purchased this product on flipkart.com. Superb Service Rather than saying on book I would Like to say about the Service that i Got by Flipkart. Superb Service, Very Fast, And Very Good Packed. This was my First Online Shopping and i m really Happy by this. Now regarding the Book, Its the Best Book For All Competitive Exams.
Calculus: An Applied Approach This accessible introduction to Calculus is designed to demonstrate how calculus applies to various fields of study. The text is packed with real ...Show synopsisThis accessible introduction to Calculus is designed to demonstrate how calculus applies to various fields of study. The text is packed with real data and real-life applications to business, economics, social and life sciences. Applications using real data enhances student motivation. Many of these applications include source lines, to show how mathematics is used in the real world. NEW! Conceptual problems ask students to put the concepts and results into their own words. These problems are marked with an icon to make them easier to assign. More opportunities for the use of graphing calculator, including screen shots and instructions, and the use of icons that clearly identify each opportunity for the use of spreadsheets or graphing calculator. Work problems appear throughout the text, giving the student the chance to immediately reinforce the concept or skill they have just learned. Chapter Reviews contain a variety of features to help synthesize the ideas of the chapter, including: Objectives Check, Important Terms and Concepts, True-False Items, Fill in the Blanks and Review Exercises. Includes Mathematical Questions from Professional Exams (CPA) Hardcover. Almost new condition. SKU: 9780471452027-2-0-3...Fine. Hardcover. Almost new condition. SKU: 97804714520271452027. Description:8th edition Our goal with every sale is customer satisfaction,...8th edition
Calculator Tutorials You do not need to go to the AfA Social Network to access these activities. Students and teachers can access these resources directly by going straight to the urls provided. Simply copy and paste the links given below and provide those links to your students. Click here for more detailed instructions. TI-84 Tutorials Tutorials for the TI-84 calculator are provided below. Each tutorial is included as a video, many being interactive, in which you must click on the appropriate buttons on the calculator to continue. To view the calculator tutorials click on the web address below. Use this address to provide access to students.Note that you can use this address to access the calculator tutorials directly without entering the AfA Social Network: In the spring of 2010 a new operating system was released for the TI-84. This operating system is a major upgrade with many useful features added. You can view an overview of the features of the new OS in the video below. Links are provided with instructions on how to upgrade your OS as well
Algebra 2 9780078738302 ISBN: 007873830X Pub Date: 2006 Publisher: Glencoe/McGraw-Hill School Pub Co Summary: THE PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! "Glencoe Algebra 2" is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics scores on today's high-stakes assessments. Holliday, Berchie is the author of Algebra 2, published 2006 under ISBN 9780078738302 and 007873830X. Five hundred ...fifty four Algebra 2 textbooks are available for sale on ValoreBooks.com, three hundred nineteen used from the cheapest price of $27.51, or buy new starting at $129.12.[read more] Ships From:Enon, OHShipping:StandardComments: 007873830X Used book with cover wear and frayed corners. This book does contains highlighting an... [more] 00THE PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! "Glencoe Algebra 2" is a key program in our vertically aligned high school mathematics series developed to help all studen [more] THE PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! "Glencoe Algebra 2" is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathemati...[
Basic Mathematics - 7th edition Summary: Patient and clear in his explanations and problems, Pat McKeague helps students develop a thorough understanding of the concepts essential to their success in mathematics. Each chapter opens with a real-world application. McKeague builds from the chapter-opening applications, such as the average amount of caffeine in different beverages, and uses the application as a common thread to introduce new concepts, making the material more accessible and engaging for student...show mores. Diagrams, charts, and graphs are emphasized to help students understand the material covered in visual form. McKeague's unique and successful EPAS system of Example, Practice, Answer, and Solution actively involves students with the material and thoroughly prepares them for working the Problem Sets. The Sixth Edition of BASIC MATHEMATICS also features a robust suite of online course management, testing, and tutorial resources for instructors and students. This includes iLrn Testing and Tutorial, vMentor live online tutoring, the Digital Video Companion CD-ROM with MathCue, a Book Companion Web Site featuring online graphing calculator resources, and The Learning Equation (TLE), powered by iLrn. TLE provides a complete courseware package, featuring a diagnostic tool that gives instructors the capability to create individualized study plans. With TLE, a cohesive, focused study plan can be put together to help each student succeed in math28496 2010 Trade paperback 7th ed. Very good. Trade paperback (US). Glued binding. 589 p. Contains: Illustrations. Audience: General/trade. I have for sale a VERY GOOD CONDITION softbound textbook o...show moref 589 pages titled " BASIC MATHEMATICS " Seventh Edition, written by charles P. McKeague with a copyright year of 2010 by Brooks/Cole ( ISBN 0-495-55974-1 ) (28496) This textbook has minor cover, corner and edge wear. A couple of cover corners are slightly dinged. A couple of small creases are visible along the bottom edge of the binding. Glimpsing
Formula10 digit dot matrix full function calculator with large color changing display and a five function clock. Features a built-in formula computation mode that supports up to an 80 character formula function. Uses 4 AAA batteries (not included).
Teaching High School Mathematics; First Course; Basic Principles for Real Numbers Part 3 Description: Max Beberman leads students on an investigation to find and prove various mathematical principles, including the Communitive Principle for Multiplication of Real Numbers, the Associative Property for Addition of Real Numbers, the Communitive Principles for Addition of Real Numbers, and the Principle for Adding the Real Number Zero. The students learn to check their findings and develop a working acquaintance with all of the principles. They also begin to use letters as variables to indicate patterns and to generate terms and sentences. In addition, Beberman introduces the pupils to using letters as indices to modify a sentence to indicate generalizations, which is discussed in future films. Black and white picture with sound. Eastman Kodak edge code reads "square triangle," which correlates to 1965.
This site has has interactive explanations and simulations of math from alegrbra to trigonometry. Just click the... see more This site has has interactive explanations and simulations of math from alegrbra to trigonometry. Just click the "interactive" tab on the top left menu and you can choose different simulations. It includes, the complete definition of parabolas, reaching beyond the ability to graph into the realm of why the graph appears as it does. It also has vivid descriptions of angles including circle angles for geometry. It also has calculators for principal nth roots, gdc, matrices, and prime factorization. It's definitely worth checking out. Quote from site: "A parabola is actually a locus of a point and a line. The point is called the focus and the line the directrix. That means that all points on a parabola are equidistant from the focus and the directrix. To change the equation and the graph of the interactive parabola below just click and drag either the point A, which is the focus, or point B, which controls the directrix." This is an interactive site that allows people to change the graph to understand why directrix and focus dictate parabolic graphs. These online notes are intended for students who are working through the textbook Abstract Algebra by Beachy and Blair. The... see more These online notes are intended for students who are working through the textbook Abstract Algebra by Beachy and Blair. The notes are focused on solved problems, and will help students learn how to do proofs as well as computations. There are also some "lab" questions on groups, based on a Java applet Groups15 written by John Wavrik of UCSD. This site contains a Java applet that provides information about groups of order 1-15. The user selects an order of a group... see more This site contains a Java applet that provides information about groups of order 1-15. The user selects an order of a group between 1 and 15 and then a model group is generated using letters as elements. Using buttons of the applet one can illustrate different algebraic properties of this group. This site contains six research projects that investigate topics in geometry and number theory. Each provides the... see more This site contains six research projects that investigate topics in geometry and number theory. Each provides the objectives, prerequisites, a summary, the project itself, and reference. The nontrivial and open-ended projects are designed to inspire students to make conjectures and research mathematics. The emphasis of each project is on obtaining results. Proofs are suggested and provided in the solutions, but students are not expected to be able to prove all the results obtained.
Description Designed for the junior/senior level of the mathematics and computer science curricula, this book teaches students how to use Mathematica to solve problems in mathematics, computer science, engineering, and other scientific fields in which mathematics and computing skills are essential. The text moves from simple to complex and often follows a specific real-world example on a number of different levels, allowing students to build their confidence and competence. The accompanying CD-ROM contains the entire text of the book as a collection of Mathematica notebooks. ContentsPart I: Learning the Basics Mathematica Basics: An Introduction \| Mathematica Basics: Help \| Mathematica Basics: Notebooks \| Mathematica Basics: Text and Typesetting \| Mathematica Basics: Packages Part II: Designing Functions Values, Variables, and Assignments \| Functions \| Recursive Definitions \| Substitution Rules and Optional Arguments \| Four Spheres Packing Problem Part III: Designing Programs List Processing Functions \| Iterations with Loops \| Computations with Modules and Local Variables Part IV: Exploring Advanced Features Advanced Mathematica: Options \| Advanced Mathematica: Hyperlinks and Buttons \| Advanced Mathematica: Packages \| Advanced Mathematica: Files, Data Exchange, and Conversions Part V: Student Projects Student Projects Related TopicsTutorial and Reference
The purpose of this book is to provide a sound introduction to the study of real-world phenomena that possess random variation. It describes how to set up and analyse models of real-life phenomena that involve elements of chance. Motivation comes from everyday experiences of probability, such as that of a dice or cards, the idea of fairness in games... more... Parametric optimization, especially in machining of non-ferrous alloys seems to be quite rare and needs an immediate attention because of its associated downstream financial and non-financial losses. This book tries to fill the gap and presents an optimization problem of commonly used Al-7020 Alloy. Principles of Response Surface Methodology (RSM)... more... The 8th edition of the successful textbook provides a compact introduction to MATLAB and its graphic extensions Simulink and Stateflow. The book also examines the most important expansion packages. Explanations are illustrated with incisive examples drawn from mathematics, physics, electrical engineering, and mechanical engineering. The reader can... more... Linear models are central to the practice of statistics and form the foundation of a vast range of statistical methodologies. Julian J. Faraway's critically acclaimed Linear Models with R examined regression and analysis of variance, demonstrated the different methods available, and showed in which situations each one applies. Following in those... more... Books on regression and the analysis of variance abound?Generalised Least Squares adopts a concise and mathematically rigorous approach. It will provide an up-to-date self-contained introduction to the unified theory of generalized least squares estimations, adopting a concise and mathematically rigorous approach. The book covers in depth the 'lower and upper bounds approach', pioneered by the first... more...
This subsite of Mathematics Tutorials and Problems (with applets) is divided into Interactive Tutorials, Calculus Problems,... see more This subsite of Mathematics Tutorials and Problems (with applets) is divided into Interactive Tutorials, Calculus Problems, and Calculus Questions, Answers and Solutions. Here the user will find applets with guided exercises and many examples and worked out problems applicable to the first year of Calculus. Quoted from the site: [This site contains...] "Free mathematics tutorials to help you explore and gain deep understanding of... see more Quoted from the site: [This site contains...] "Free mathematics tutorials to help you explore and gain deep understanding of math topics." The math topics covered include 1) Precalculus Tutorials 2) Calculus Tutorials and Problems 3) Geometry Tutorials and Problems 3) Trigonometry Tutorials and Problems for Self Tests 4) Elementary statistics and probability tutorials 5) Applications of mathematics in physics and engineering. And much more, including many applets. The sculptures by Brent Collins was an eye catch with the different shapes, this was the first that I have see of this type... see more The sculptures by Brent Collins was an eye catch with the different shapes, this was the first that I have see of this type of math. I also like the utility curve for mathematics - Bit-Player by Brian Hayes – the curve for life. It's like the title of our booklet "MATH IS EVERYWHERE". This site offers assistance for all types of math from basic math through calculus. There are text and video lessons as well... see more This site offers assistance for all types of math from basic math through calculus. There are text and video lessons as well as practice tests. These tutorials are based on a large variety of textbooks available from middle school through post-secondary
general survey of mathematical topics helps a diverse audience, with different backgrounds and career plans, to understand mathematics. Blitzer provides the applications and technology readers need to gain an appreciation of mathematics in everyday life. Demonstrates how mathematics can be applied to readersrs" lives in interesting, enjoyable, and meaningful ways. Features abundant, step-by-step, annotated Examplesthat provide a problem-solving approach to reach the solution; annotations are conversational in tone, explaining key steps and ideas as the problem is solved. Begins each section with a compelling vignette highlighting an everyday scenario, posing a question about it, and exploring how the chapter section subject can be applied to answer the question. A highly readable reference for anyone who needs to brush up their mathematics skills.
Calculus Course Outline and Objectives Unit One: Prerequisites A brief Review of: A. The set of Real Numbers, inequalities, absolute value and its properties. B. The distance and midpoint formulas, C. Graphing circles. D. Graphing equations with and without a graphics calculator, locating x and y-intercepts, identifying the symmetries associated with a given equation, locating the points of intersection of two equation by algebra and with the use of the graphics calculator. E. Finding the equation of a line given: 1. A point and a slope. 2. Two points. 3. That is parallel or perpendicular to a given line that passes through a given point. F. Functions 1. Find the domain and range of a given function. 2. Evaluate and create composite functions. 3. Determine whether a given function is odd, even, or neither. G. Trig functions 1. Graphing trig functions using a graphics calculator. 2. Work with conversions from degrees to radians and radians to degrees and with problems that involve arc length. 3. Solve trig equations and work with trig identities. 4. Evaluate trig functions. Unit Two: A Detailed Study of Limits & Their Properties A. Formal definition of limit 1. Exploring limits that exist vs. limits that do not exist B. A detailed study of the limit properties 1. Scalar multiplication property 2. Sum and difference properties 3. Product and quotient properties 4. Power property C. Exploring techniques for evaluating limits 1. Making tables of values using the graphics calculator 2. Simplifying using algebra and rationalization skills D. Exploring one-sided limits 1. A study of the greatest integer function, the absolute value function, and rational functions. E. Infinite limits 1. A study of continuity on an open interval 2. A study of removable and non-removable points of discontinuity. 3. A study of vertical asymptotes. F. Demonstrate the use of the Intermediate Value Theorem. Unit Three: A Detailed Study of Differentiation A. Using the limit definition to find the slope of a tangent line to a curve at a specific point on the curve, and then use this slope to find the equation of the tangent line. B. Determine whether a function is differentiable at a given point by checking for: 1. Continuity at the given point. 2. A vertical tangent line at the given point 3. A sharp cusp in the graph. C. Learn to use the following rules to differentiate given functions. 1. The derivative of a constant is zero. 2. The Power Rule 3. The sum and difference rules. 4. The product rule. 1 5. The quotient rule. 6. The Chain Rule. D. Learn how to differentiate expressions that contain trig functions. E. Use implicit differentiation to calculate derivatives. 1. Apply knowledge of implicit differentiation to solving rate of change problems. F. Apply differentiation rules to taking higher order derivatives. G. Solving problems using know of derivatives. 1. Velocity and acceleration problems. 2. Rate of change problems. 3. Projectile motion problems Unit Four: Applications of Derivatives A. Finding the extrema of a function 1. Locate all critical values on a closed interval. a. Interval endpoints b. Where f (x) = 0 or where f (x) is undefined. 2. Evaluate the critical values to identity the extrema (maximum and minimum) on a open or closed interval. B. Explore the use of Rolles Theorem and the Mean Value Theorem. C. Explore the use of the first derivative test. 1. To identify pause points and use a line test to determine if the point is a local maximum or minimum points. 2. To determine whether a function is increasing or decreasing in a given interval. D. Explore the concavity using the second derivative test. E. Using knowledge gained from the first and second derivative tests to sketch accurate graphs of functions. F. Continued study of limits that approach infinity to identify vertical and horizontal asymptotes to aid in graphing functions. G. Introduction to the study of differential equations. H. A detailed study of optimization problems. I. An introduction to the study of business and economic applications that involve the use of derivatives. Unit Five: Detailed Study of Integration A. Introduction to various notations used to denote antiderivatives, the rules that apply to taking antiderivatives, introduction to indefinite integrals, and to finding a particular solution to an indefinite integral. B. Introduction to sigma notation and its use to find areas under curves using the limit definition. C. Introduction to the use of Riemann Sums and their use to evaluating definite integrals. D. Introduction to the Fundamental Theorem of Calculus and its use to evaluate definite integrals. E. Evaluating definite integrals using substitution techniques. Unit Six: A Detailed Study of Logarithmic & Exponential Differentiation and Integration A. Review of log properties and their use to solve equations. B. Applying rules for taking derivatives to functions that include ex, ln(x), log(x) and ax. C. Continued study of implicit differentiation. D. Introduction to logarithmic differentiation. E. Applying the rules for integrating define and indefinite integrals to integrating expressions that include ex, ax, and (1/x). F. Solving word problems that deal with exponential growth and decay. Unit Seven: Calculating the Area Between Curves & Finding Volumes of Revolution A. Calculating the area between a curve and either the x-axis or the y-axis using given parameters for the variables. Calculating the area between two curves. B. Calculating the volume of a region that is revolved about the x-axis, the y-axis, or a horizontal or vertical line using disks, washers, and shells
More About This Textbook Overview Essential Mathematics for Games and Interactive Applications, 2nd edition presents the core mathematics necessary for sophisticated 3D graphics and interactive physical simulations. The book begins with linear algebra and matrix multiplication and expands on this foundation to cover such topics as color and lighting, interpolation, animation and basic game physics. Essential Mathematics focuses on the issues of 3D game development important to programmers and includes optimization guidance throughout. The new edition Windows code will now use Visual Studio.NET. There will also be DirectX support provided, along with OpenGL - due to its cross-platform nature. Programmers will find more concrete examples included in this edition, as well as additional information on tuning, optimization and robustness. The book has a companion CD-ROM with exercises and a test bank for the academic secondary market, and for main market: code examples built around a shared code base, including a math library covering all the topics presented in the book, a core vector/matrix math engine, and libraries to support basic 3D rendering and interaction. Editorial Reviews From the Publisher Product Details ISBN-13: 9780123742971 Publisher: Taylor & Francis Publication date: 5/19/2008 Edition description: New Edition Edition number: 2 Pages: 704 Product dimensions: 7.60 (w) x 9.30 (h) x 1.70 (d) Meet the Author James M. Van Verth is a founding member of Red Storm Entertainment, a division of Ubisoft, where he has been a lead engineer for nine years. For the past eight years he has also been a regular speaker at the Game Developers Conferences, teaching the all-day tutorial "Math for Game Programmers," on which this book is based. He has a B.A. in mathematics and computer science from Dartmouth College and M.S. degrees in computer science from the State University of New York at Buffalo and the University of North Carolina at Chapel Hill. Lars M. Bishop is an engineer in the Handheld Developer Technologies group at NVIDIA. Prior to joining NVIDIA, Lars was the Chief Technology Officer at Numerical Design Limited, leading the development of the Gamebryo3D cross-platform game engine. He received a BS in Math/Computer Science from Brown University and an MS in Computer Science from the University of North Carolina at Chapel Hill. His outside interests include photography, drumming, and playing bass guitar. Table of Contents The following gives a high-level table of contents. Starred chapters will change significantly from the current edition, with changes noted. Chapter 1: Vectors and Points This chapter covers basic linear algebra topics such as vectors, points, lines and planes. Most changes will be in presentation, not in topic. Chapter 2: Matrices and Linear Transformations This continues the linear algebra discussion and discusses the mathematical methods for changing the concepts presented in Chapter 1. Most modifications will be in presentation, not in topic. Chapter 3: Affine Transformations This completes the algebra discussion and covers basic graphics transformations such as translation, rotation and scale. Most changes will be in presentation, not in topic, although there will be a small section added on skinning. Chapter 4: Real-World Computer Number Representation This chapter covers the problems of representing the infinite space of numbers in a finite computer. Most modifications will be in presentation, not in topic. Chapter 5: Viewing and Projection This chapter begins the graphics topics with a discussion of building virtual cameras. Most modifications will be in presentation, not in topic. Chapter 6: Lighting and Color This new chapter collates the color topics from the original Chapter 6 and the non-shading aspects of the original Chapter 7. To be presented here are the more abstract notions of color, normals to a surface, direction to a light, and how a lit color value is generated without any reference to triangles or interpolation. Chapter 7: Geometry, Shading and Texturing As with the previous chapter, this one will take mostly material from the original chapters 6 and 7, and some from Chapter 8. The chapter will conclude with a brief discussion of hardware vertex and pixel shaders, but as this is not a graphics book per se it will only be at an introductory level. Chapter 8: Rasterization and Antialiasing The final graphics chapter will now focus on sampling the continuous functions discussed in the previous two so that they can be represented on discrete hardware. Both rasterization and texturing will lead into a discussion of antialiasing and solving the problems for both processes: mip-mapping, multi-sampling, and other approaches. Chapter 9: Animation of Position The previous version of this chapter focused on curves, and the new one will be no exception. However, rather than attempting to be a general curve discussion it will focus more on the particular curve types that are most often used in an interactive application for animating cameras and object position. Chapter 10: Animation of Orientation As with the previous chapter, this one will be retargeted to focus mainly on animation issues. There will also be more material dedicated to approximating slerp, as there have been a number of publications and debate on this topic since the original publication of the book. Chapter 11: Intersection Testing This chapter covers basic bounding object tests and some topics in collision detection and culling. There will be few changes. Chapter 12: Physical Simulation The general structure of this one will remain the same with only a few additions and expansions of existing topics based on the author's participation in the Physics for Game Programmers GDC tutorial. First, symplectic Euler and its stability properties will be discussed. Secondly, more attention will be given to generating inertial tensors for triangle soups. Finally, simple constraints will be presented. Chapter 13: Random Numbers This chapter was originally planned for the first edition and was requested by a number of reviewers. It will cover basic probability, three methods for generating pseudo-random numbers on a digital computer, and some applications of random numbers: Artificial intelligence routines, simulating die rolls and card draws, Perlin noise for procedural textures, and finally some basic presentation of stochastic sampling for anti-aliasing
A Course in Computational Number Theory A Course in Computational Number Theory is an introduction to number theory, with emphasis on a broad range of computational issues. Mathematica files (on CD-ROM) with all the commands and programs are included as a tool for motivation and explanation. This book covers the theoretical structure of number theory, but expands with the software into exploration of many topics including factoring algorithms, primality testing, and the RSA public-key cryptosystem. Unusual and advanced topics also covered include check-digit schemes, continued fractions, Pell's equation, and the Gaussian primes.
Don;t want an expensive calculator with functions you would never use?Do you want to solve your trigonometric problems in a... see more 'Don;t want an expensive calculator with functions you would never use?Do you want to solve your trigonometric problems in a faster and more pleasing way?Well then , Designer Trigonometric Calculator is the right thing for you!With al the trigonometric functions in just one application , you'll be solving your problems in no time.!!!Choose the way you like your calculator to be!Calculate Sin , Cos , Tan etc. in degrees or radians as per your wish in this packaged application!'This is a free app A new curriculum, and then a book based upon that, always seeks evaluation. The book should not only be attractive and... see more A new curriculum, and then a book based upon that, always seeks evaluation. The book should not only be attractive and interesting, but also upto the mark with the age level of students, appropriateness of content, relevance, utility, etc. Every new book should be thoroughly evaluated before it is given a place in list of textbook or suggestive readings or after introducing in a course. An exercise for evaluating a textbook has been presented in the current book. For this purpose, a textbook for class IX Mathematics, introduced in schools, has been selected. The study goes through explaining various aspects of evaluating the book. It makes an effort to see the book through the lenses of teachers as well as students. The book looks forward to place teachers not just as mere followers of the book introduced, but as reflectors, too, upon various aspects of the book. In this book, the entire study consisted of basically five phases. In the present investigation, the definition of... see more In this book, the entire study consisted of basically five phases. In the present investigation, the definition of standardization has been adopted as given by Roger T. Lennon i.e. here standardization means to determine validity, reliability and norms of the test. The definition is also mentioned in "Measurement in Physical Education" by Donald K. Mathews (1978) The five phases are as follows: Phase I: Planning and preparation of a test set of 50 questions from Trigonometry for class IX. Phase II: Percentage of agreeability or content-cum-face validity with the help of subject experts and field experts. Phase III: Group try out for Item analysis. Phase IV: The final try-out to obtain reliability Phase V: Developing norms for the test. The above design of the study is also supported by Lee J. Cronbach (1984). 'Taking discrete mathematics? Then you need the Wolfram Discrete Mathematics Course Assistant. This app for discrete... see more 'Taking discrete mathematics? Then you need the Wolfram Discrete Mathematics Course Assistant. This app for discrete math-- -'This app costs $4.99
Elementary Numerical Analysis 9780471433378 ISBN: 0471433373 Edition: 3 Pub Date: 2003 Publisher: Wiley Summary: Offering a clear, precise, and accessible presentation, complete with MATLAB programs, this new Third Edition of Elementary Numerical Analysis gives students the support they need to master basic numerical analysis and scientific computing. Now updated and revised, this significant revision features reorganized and rewritten content, as well as some new additional examples and problems. The text introduces core areas... of numerical analysis and scientific computing along with basic themes of numerical analysis such as the approximation of problems by simpler methods, the construction of algorithms, iteration methods, error analysis, stability, asymptotic error formulas, and the effects of machine arithmetic. Kendall Atkinson is the author of Elementary Numerical Analysis, published 2003 under ISBN 9780471433378 and 0471433373. Six hundred ninety six Elementary Numerical Analysis textbooks are available for sale on ValoreBooks.com, one hundred nineteen used from the cheapest price of $65.99, or buy new starting at $125.500471433373 Brand new book. Hardcover US edition. Ship from multiple locations, including USA, UK, Asia. 3-5 business days Express Delivery to USA/UK/Europe/Asia/Worldwide. Tracking number will be provided. Satisfaction guaranteed. ISBN: 0471433373
This applet explores Riemann sums including left endpoint, right endpoint, midpoint, inscribed rectangle, circumscribed... see more This applet explores Riemann sums including left endpoint, right endpoint, midpoint, inscribed rectangle, circumscribed rectangle, and trapezoid. The visual representation of the partitions and the corresponding value of the sum are provided. The "fourth edition" of a free, on-line programming textbook,directed towards people who are new to programming.It is... see more The "fourth edition" of a free, on-line programming textbook,directed towards people who are new to programming.It is available for on-line use and for downloading. Includes160 source code files and 65 solved programming exercises.This version requires Java 1.3 or higher 'Foundations of Computation is a free textbook for a one-semester course in theoretical computer science. It has been used... see more 'Foundations of Computation is a free textbook for a one-semester course in theoretical computer science. It has been used for several years in a course at Hobart and William Smith Colleges. The course has no prerequisites other than introductory computer programming. The first half of the course covers material on logic, sets, and functions that would often be taught in a course in discrete mathematics. The second part covers material on automata, formal languages, and grammar that would ordinarily be encountered in an upper level course in theoretical computer science.Version 2.3 (Summer 2010) added a section on pushdown automata; aside from that, there were only minor corrections and changes. The most recent version, 2.3.1 (Summer 2011), is a very minor update, with one new proof and a few corrections.Table of Contents:Chapter 1: Logic and ProofChapter 2: Sets, Functions, and RelationsChapter 3: Regular Expressions and FSA'sChapter 4: GrammarsChapter 5: Turing Machines and Computability' A pentomio consists of 5 connected squares. There are 12 possible pentominos. The puzzle is to place them on an 8-by-8 board,... see more A pentomio consists of 5 connected squares. There are 12 possible pentominos. The puzzle is to place them on an 8-by-8 board, with 4 designated squares left blank. This applet solves the puzzle (using a recursive backtracking algorithm that runs in its own thread). The xFunctions applet is designed as an educational tool for exploring several topics in calculus and pre-calculus... see more The xFunctions applet is designed as an educational tool for exploring several topics in calculus and pre-calculus mathematics. In addition to basic graphing, xFunctions can create animated graphs, parametric curves, derivatives, Riemann sums, integral curves, and 3D graphs. Helps you to learn the basic concepts of hierarchical, geometrical modeling in computer graphics and animations. This applet... see more Helps you to learn the basic concepts of hierarchical, geometrical modeling in computer graphics and animations. This applet takes descriptions of images and animations and displays them in wireframe form.
College Algebra : Graphs and Models -Text Only - 4th edition Summary: The approach of this text is more interactive than most precalculus texts and the goal of the author team is to enhance the learning process through the use of technology and to provide as much support and help for students as possible. In Connecting the Concepts, comprehension is streamlined and retention is maximized when the student views a concept in visual, rather than paragraph, form. Zeros, Solutions, and X-Intercepts Theme Carried Throughout helps students vi...show moresualize and connect the following three concepts when they are solving problems: the real zeros of the function, the solutions of the associated equation, and the x-coordinates of the x-intercept of the graph of the function. Each chapter begins with a relevant application highlighting how concepts presented in the chapter can be put to use in the real world. These applications are accompanied by numerical tables, equations, and grapher windows to show students the many different ways in which problems can be examined. End-of-Chapter material includes a summary and review of properties and formulas along with a complete set of review exercises. Review exercises also include synthesis, critical thinking, and writing exercises. The answers to all of the review exercises appear in the back of the text and have text section references to further aid students. For anyone interested in learning algebra
How do we know that what we know is, in fact, true? And how do we communicate what we know so that others will be convinced that it is true? These questions are central to the idea of mathematical proof and to the motivating force for teaching students of mathematics about proof and how to do proof. The ability to communicate mathematically, more generally, the ability to communicate logically, forcefully and convincingly is an invaluable skill to which all students should be exposed. Mathematics is an excellent vehicle for teaching students this vital skill. The curriculum unit described in this document is designed to lead students to an understanding of what proof is, why it is important and how to construct good proofs. Essentially it is a unit about argument and communication and in this way crosses curricular lines by promoting skills which are universal and useful in the sciences, social sciences and humanities. Most would agree that to possess common sense is a good thing. As a science educator, however, I have found that common sense can be inadequate in explaining or in understanding of many topics. This paper will examine instances where common sense can be misleading and an inadequate explanatory tool. Explanations and proofs that go beyond common sense are presented to explain topics. Lessons and instructional strategies that support the explanations are suggested that are suitable for middle school students. Topics covered include freely falling bodies, determining the shape and circumference of the earth, floating and sinking, vacuums, and logic and mathematic problems. In every case, the topic will be thoroughly discussed and analyzed and, hopefully, interested teachers can find useful information for their own classes.
More About This Textbook Overview ELEMENTARY STATISTICS: A BRIEF VERSION is for introductory statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. In recent editions, Al Bluman has placed more emphasis on conceptual understanding and understanding results, along with increased focus on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing technologies commonly used in such courses. The 6th edition of Bluman, Elementary Statistics: A Brief Version provides a significant leap forward in terms of online course management with McGraw-Hill's homework platform, Connect Statistics – Hosted by ALEKS. Statistic instructors served as digital contributors to choose the problems that will be available, authoring each algorithm and providing stepped out solutions that go into great detail and are focused on areas where students commonly make mistakes. From there, the ALEKS Corporation reviewed each algorithm to ensure accuracy. The result is an online homework platform that provides superior content and feedback, allowing students to effectively learn the material being
edshelf Algebra Genie Description Ages of learners Platforms Categories Subjects Next-generation interactive textbook, at the fraction of the cost of tutoring or your current paper textbook! The most complete and interactive Algebra course, developed by over 100 Algebra teachers and used by millions of Algebra students worldwide. The Algebra Genie gets you learning quickly while having fun. Developed using the latest in game mechanics, we have spared no cost to make sure we have the best learning app covering Algebra! Over 250 interactive and dynamic lessons, covering all the important Algebra topics, to get you into college quickly! Topics covered: Algebraic Expressions Exponents Linear Relations The Pythagorean Theorem Function Basics Functions Quadratic Functions Absolute Function Square Root Function Step Functions Exponentials & Logarithms Factoring Systems of Equations Conics Stay tuned for our other apps coming to the App Store, such as Geometry and Trigonometry.
Product Description Product Description Maths isn't dead – it just smells funny. Here's a professional tutor and former-rocket-scientist, making EdExcel's Core 1 Mathematics module understandable. Clear, well-thought out answers and explanations, designed to help you get this module understood and out of the way ASAP. This book covers: - Surds and powers - Dealing with fractions - Factorising quadratics and completing the square - Real roots and the discriminant - Inequalities - Arithmetic series - Equations of a line - Triangles and distances - Sketching graphs and transformations - Simultaneous equations - Differentiation - Integration - When to differentiate and when to integrate - Co-ordinate geometry (the tangenty normally questions) More About the Author Colin Beveridge (1977-) is a maths confidence coach for Flying Colours Maths and co-author of the Little Algebra Book. He holds a PhD in Mathematics from the University of St Andrews and worked for several years on NASA's Living With A Star project at Montana State University, where he came up with an equation which is named after him. It's used to help save the world from being destroyed by solar flares. So far so good. He became tired of the glamour of academia and returned to the UK to concentrate on helping students come to terms with maths and show that not all mathematicians are boring nerds; some are exciting, relatively well-adjusted nerds. Colin lives in Dorset with an espresso pot, several guitars and nothing to prove. book is a revision/help guide for pupils studying Edexcel A Level Maths C1. It is written in a way which takes students logically through the sticking points of C1 and gives plenty of help on the understanding of the topic, and plenty of little tricks and mnemonics to help them remember stuff in the exam. The book explains things brilliantly, and in a way that is humorous and engaging. I don't think I've ever laughed out loud at a revision guide before! My only slight criticisms of the book are that on a few occasions, i.e. transforming graphs and dividing fractions, the help is constrained to the "how" of it, with not the why. And that Colin dismisses completing the square as pointless, but I feel that it is great for explaining the quadratic formula, is useful for seeing if a graph has real solutions and has its uses for higher level polynomials for those who go past A Level. Colin and I differ on our opinions on the best way to complete the square, but his explanation of the equating co-efficients method is superb. If you, or someone you know, is struggling with any of the topics involved in C1 then I would certainly advise buying a copy for your kindle. It is far better than any of the other C1 revision guides I have come across, and would be worth the money for the jokes, or the story of "bad guy x", alone!
0141030232 9780141030234 The Penguin Dictionary of Mathematics:Covering all branches of pure and applied mathematics, including algebra, geometry, mechanics, and statistics, The Penguin Dictionary of Mathematics is invaluable for students at the high school or university level. It is also a useful and versatile source book for economists, business people, engineers, technicians, scientists of all kinds, and anyone else who uses mathematics in the course of daily work. Back to top Rent The Penguin Dictionary of Mathematics 4th edition today, or search our site for David textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Penguin Books.
This well-prepared text on category theory is intended for beginners with an elementary knowledge of set theory. The first five chapters are devoted to the fundamentals of category theory up to adjoints and monads. The remaining two chapters deal with more recent research results in the realm of topological and algebraic categories, Cartesian closed categories and quasitopoi. The text prefers a pedagogical style, introducing new concepts first in special cases and then moving to the more general ones. Each categorical notion is always accompanied by many examples from diverse branches.
Grace be upon god,with blessing,this Additional Mathematics project work finally have been done. Firstly,I would like to express my appreciation to my Additional Mathematics teacher Puan Asmalia for her contructive critism and helpful suggestion during the process to finish this Additional Mathematics project work. Not to forget,my parents that also give me a lot of help and supporting me to complete this project.Besides that,I also wish to thank to the member of my group for their support to make sure that this project work can be finished. Lastly,I also want to thank to whom that participate in the process to finish this project work wether indirect or not. 1 Objective : Theaimsof carrying out this project work are to enable students to : Apply mathematics to everyday situations and appreciate the importance and the beauty of mathematics in everyday lives Develop mathematical knowledgein a way which increasestudents' interest and confidence. 2 Method investigation In solving and finishing this project work done,some method is used :- 1.Communication Discussion with teacher and friend help in solving problem.The information from this discussion used as a reference materials to success this project. 2.Reference Additional of information from various of reference material help me to find the method to solve the problem.For this Additional Mathematics project,I can get the reference from library,internet,my friends,my teacher and many more. 3.Lesson session The lesson session in the class help me in solve problem by using heuristics what I learn in the class. 3 Introduction. The Petronas Towers of Kuala Lumpur Often we know the relationship involving the rate of change of two variables, but we may need to know the direct relationship between the two variables. For example, we may know the velocity of an object at a particular time, but we may want to know the position of the object at that time. To find this direct relationship, we need to use the process which is opposite to differentiation. This is called integration (or antidifferentiation). The processes of integration are used in many applications. The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the building for strength. 4 Sydney Opera House The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building. Wine cask Historically, one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface).Here is a LiveMath illustration of a 3D shape. We learn to find the volume of these objects later (in volume of solid of revolution). Other uses of integration include finding areas under curved surfaces, centres of mass, displacement and velocity, fluid flow, modelling the behaviour of objects under stress, etc. 5 Introduction of integration Inmathematics,integration is a technique of finding a functiong(x) the derivativeof which,Dg(x), is equal to a given functionf(x). This is indicated by the integral sign "∫," as in ∫f(x), usually called theindefinite integralof the function. (The symboldxis usuallyadded, which merely identifiesxas the variable.) Thedefinite integral, written withaandbcalled the limits of integration, is equal tog(b) −g(a), whereDg(x) =f(x).Some antiderivatives can be calculated by merely recalling which function has a given derivative, but the techniques of integration mostly involve classifying the functions according to which types of manipulations will change the function into a form the antiderivative of which can be more easily recognized. For example, if one is familiar with derivatives, the function 1/(x+ 1) can be easily recognized as the derivative of loge(x+ 1). The antiderivative of (x2+x+ 1)/(x+ 1) cannot be so easily recognized, but if written 6 As x(x+ 1)/(x+ 1) + 1/(x+ 1) =x+ 1/(x+ 1), it then can be recognized as the derivative of x2/2 + loge(x+ 1). One useful aid for integration is the theorem known asintegration by parts. In symbols,the ruleis ∫fDg=fg− ∫gDf.That is, if a function isthe productof two other functions,fand one that can be recognized as the derivative of some functiong, then the original problem can be solved if one can integratethe productgDf.For example, iff=x, andDg= cosx, then ∫x·cosx=x·sinx− ∫sinx=x·sinx− cosx+C. Integrals are used to evaluate such quantities as area, volume, work, and, in general, any quantity that can be interpreted as the area under a curve. 7 Definition The process of finding a function, given its derivative, is calledanti-differentiation(orintegration). IfF'(x) =f(x), we sayF(x)is an anti-derivative off(x). Examples Note that ifF(x)is an anti-derivative off(x)thenF(x) +c, wherecis a constant (called theconstant of integration) is also an anti-derivative ofF(x), as the derivative of a constant function is 0. In fact they are the only anti-derivatives ofF(x). We writef(x)dx=F(x) +c. ifF'(x) =f(x). We call this the indefinite integral off(x). Thus in order to find theindefinite integralof a function, you need to be familiar with the techniques ofdifferentiation. 8 HISTORY Over 2000 years ago, Archimedes(287-212 BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the paraboloid. His method of integration was remarkably modern considering that he did not have algebra, the function concept, or even the decimal representation of numbers. Leibniz (1646-1716) and Newton (1642-1727) independently discovered calculus. Their key idea was that differentiation and integration undo each other. Using this symbolic connection, they were able to solve an enormous number of important problems in mathematics, physics, and astronomy. Fourier (1768-1830) studied heat conduction with a series of trigonometric terms to represent functions. Fourier series and integral transforms have applications today in fields as far apart as medicine, linguistics, and music. 9 Gauss (1777-1855) made the first table of integrals, and with many others continued to apply integrals in the mathematical and physical sciences. Cauchy (1789-1857) took integrals to the complex domain.Riemann(18261866)andLebesgue(18751941) put definite integration on a firm logical foundation. Liouville (1809-1882) created a framework for constructive integration by finding out when indefinite integrals of elementary functions are again elementary functions. Hermite (1822-1901) found an algorithm for integrating rational functions. In the 1940s Ostrowski extended this algorithm to rational expressions involving the logarithm. In the 20th century before computers, mathematicians developed the theory of integration and applied it to write tables of integrals and integral transforms. Among these mathematicians were Watson, Titchmarsh, Barnes, Mellin, Meijer, Grobner, Hofreiter, Erdelyi, Lewin, Luke, Magnus, Apelblat, Oberhettinger, Gradshteyn, Ryzhik, Exton, Srivastava, Prudnikov, Brychkov, and Marichev. 10 In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary functions. His algorithm does not automatically apply to all classes of elementary functions because at the heart of it there is a hard differential equation that needs to be solved. Efforts since then have been directed at handling this equation algorithmically for various sets of elementary functions. These efforts have led to an increasingly complete algorithmization of the Risch scheme. In the 1980s some progress was also made in extending his method to certain classes of special functions. The capability for definite integration gained substantial power inMathematica, first released in 1988. Comprehensiveness and accuracy have been given strong consideration in the development ofMathematicaand have been successfully accomplished in its integration code. Besides being able to replicate most of the results from well-known collections of integrals (and to find scores of mistakes and typographical errors in them),Mathematicamakes it possible to calculate countless new integrals not included in any published handbook. 11 12 Part 1 Route 1.1 2.1 1.2 2.2 1.3 2.3 Distnace 131km 24km √109km √307km √85km √104km Bearing Goes to north Goes to east N73.3 º N27.9ºW N77.5ºE N78.7ºE Coordinates Possible Dangers Coral reef Shark, infested water Coral, reef, sunken ship Shark, infested water, sunken ship, thunderstorm Giant octopus Giant octopus, thunderstorm Time For route 1(1.1,1.2,1.3)=55minutes 59seconds For route 2(2.1,2.2,2.3)=1hour 31minutes36seconds Judging from the possible dangers & possibilities of intruding into the preserved and conservation aresas and the time taken to reach the offshore oil rig,route 1 is the recommended option. 13 Part 2 Starting position Vresultant=Vboat+Vcurrent =+ Vresultant= = = Vcurrent= v=36sin a V=60sin a _________① v=36cos a-15______② From ①, ②,we get a=22.4º,v=22.55km/h Time taken=hour =0.4375hour 14 From Vresultant= = Vboat= Vcurrent= Vresultant=Vboat+Vcurrent =+ By using the similar concept as shown in step ①,B=54.6 º,v=29.915km/h Time taken=hour=20.3416 15 From Vresultant=Vboat+Vcurrent Vcurrent= Vboat= Vresultant= = Similary,by working it out youself,C=20.3 º,v=22.548km/h Time taken= =0.48hour Time to reach the wind – farm=10.00a.m+26minutes15seconds+20minutes28seconds =10:46:43+2hours+28minutes48seconds =13:15:31a.m 16 Part 3 P=cAu2 C= = = =5.917 b)(1)E= 50000000= = =10000t t=5000seconds (2)500000000= = = = = t=1850.6seconds 17 Part 4 a)v=πR2h = Vfull=1000000000 = πR2h πR2(3000)=100000000 πR2= v= πR2h = =__________① 3000metres=(10x365x24)hours = = = =barrels per hour 18 b) V=πr2h =π(0.25)2h =0.0625πh =0.0625π_______① Vfull=π(0.25)2(1) =0.0625π Tfull=(5x60)seconds =________② = x = 0.0625π = =20cms-1 19 Part 5 Oil Production & Consumption, Top 20 Nations by Production (% of Global) Here are the top 20 nations sorted by production, and their production and consumption figures. Saudi Arabia produces the most at 8,711,000.00 bbl per day, and the United States consumes the most at 19,650,000.00 bbl per day, a full 25% of theworld'soil consumption. Exports & Imports Here's export and imports for all the nations listed in the CIA World Factbook, sorted alphabetically as having exports and imports. Conspicuously missing is the United States, but I can tell you that we consume 19,650,000.00 bbl per day, and produce 8,054,000.00, leaving a discrepancy of 11,596,000.00 bbl per day. This compares to the European Union, which produces 3,244,000.00 bbl per day and consumes 14,480,000.00 bbl per day for a discrepancy of 11,236,000.00 per day. Basically, about the same. World Oil Market and Oil Price Chronologies: 1970 – 2003. Oil Reserves - Top 20 Nations (% of Global) Saudi Arabia has 261,700,000,000 barrels (bbl) of oil, fully 25% of the world's oil. The United States has 22,450,000,000 bbl. The United States government recently declared Alberta's oil sands to be 'proven oil reserves.' Consequently, the U.S. upgraded its global oil estimates for Canada from five billions to 175 billion barrels. Only Saudi Arabia has more oil. The U.S. ambassador to Canada has said the United States needs this energy supply and has called for a more streamlined regulatory process to encourage investment and facilitate development. - CBC Television - the nature of things - when is enough enough Further Exploration Petroleum engineers work in the technical profession that involves extracting oil in increasinglydifficult situations as the world's oil fields are found and depleted. Petroleum engineers searchthe world for reservoirs containing oil or natural gas. Once these resources are discovered, petroleum engineers work with geologists and other specialists to understand the geologicformation and properties of the rock containing the reservoir, determine the drilling methods to be used, and monitor drilling and production operations Low-end Salary: $58,600/yr Median Salary: $108,910/yr High-end Salary: $150,310/yr EDUCATION: Engineers typically enter the occupation with a bachelor¶s degree in mathematics or anengineering specialty, but some basic research positions may require a graduate degree. Mostengineering programs involve a concentration of study in an engineering specialty, along withcourses in both mathematics and the physical and life sciences. Engineers offering their servicesdirectly to the public must be licensed. Continuing education to keep current with rapidlychanging technology is important for engineers. MATH REQUIRED: College AlgebraGeometryTrigonometryCalculus I and IILinear AlgebraDifferential EquationsStatistics WHEN MATH IS USED: Improvements in mathematical computer modeling, materials and the application of statistics, probability analysis, and new technologies like horizontal drilling and enhanced oil recovery,have drastically improved the toolbox of the petroleum engineer in recent decades. POTENTIAL EMPLOYERS: About 37 percent of engineering jobs are found in manufacturing industries and another 28 percent in professional, scientific, and technical services, primarily in architectural, engineering,and related services. Many engineers also work in the construction, telecommunications, andwholesale trade industries. Some engineers also work for Federal, State, and local governmentsin highway and public works departments. Ultimately, the type of engineer determines the typeof potential employer. FACTS: Engineering diplomas accounted for 12 of the 15 top-paying majors, with petroleum engineeringearning the highest average starting salary of $83,121. Conclusion I have done many researches throughout the internet anddiscussing with a friend who have helped me a lot in completing this project. Through the completion of this project, I have learned many skills and techniques. This project really helps me to understand more about the uses of progressions in our daily life. This project also helped expose the techniques of application of additional mathematics in real life situations. While conducting this project, a lot of information that I found.Apart from that, this project encourages the student to work together and share their knowledge. It is also encourage student to gather information from the internet, improve thinking skills and promote effective mathematical communication.Last but not least, I proposed this project should be continue because it brings a lot of moral values to the student and also test the students understanding in Additional Mathematics. 27 Reflection After spending countless hours,day and night to finish this Additional Mathematics Project,here is what I got to say: Doing this project makes me realize how important Additional Mathematics is.Also, completing this project makes me realize how fun it is and likable is Additional Mathematics
and Trigonometry: A Graphing Approach Part of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, Algebra and Trigonometry: A Graphing Approach, 5/e, is an ...Show synopsisPart of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, Algebra and Trigonometry: A Graphing Approach, 5/e, is an ideal student and instructor resource for courses that require the use of a graphing calculator. The quality and quantity of the exercises, combined with interesting applications and innovative resources, make teaching easier and help students succeed. Continuing the series' emphasis on student support, the Fifth Edition introduces Prerequisite Skills Review. For selected examples throughout the text, the Prerequisite Skills Review directs students to previous sections in the text to review concepts and skills needed to master the material at hand. In addition, prerequisite skills review exercises in Eduspace (see below for description) are referenced in every exercise set. The Larson team achieves accessibility through careful writing and design, including examples with detailed solutions that begin and end on the same page, which maximizes the readability of the text. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles
Arfken is really bad to learn from. However, it's a text I wish I could simply upload into my brain. It's very dense and the #1 general resource on my bookshelf. Whenever I need to recall some sort of mathematical procedure or area, I go to Arfken. Most of the time it will be all I need and only when I need a very in-depth treatment or obscure concept do I go to a more specialized text. I agree with this. Another more modern text, at about the level of Arfken and Weber is Hassani which is much better to learn from in my experience.
Common Core Exemplar for High School Math You are here This exemplar has been developed to guide high school students and instructors with Common Core standards for Math. Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is a result of applying operations to simpler expressions. Viewing an expression as the result of operations on simpler expressions can sometimes clarify its underlying structure. This high school exemplar is intended to extend and connect students' understandings from middle school to polynomials. It focuses on reinforcing knowledge and skills, rather than developing completely new knowledge and skills.
Did you like Math or Science in school? Have you played games that stimulated your thought processes for Math and Science? Trying to be creative in your Math, Science or leadership class? Can leadership be taught? Is leadership an Art or a Science or Math? Seeking to impact your training program with creative games?. A primer for leadership development,... more... Founded in 1931 by Otto Neugebauer as the printed documentation service "Zentralblatt fur Mathematik und ihre Grenzgebiete", Zentralblatt MATH (ZBMATH) celebrates its 80th anniversary in 2011. Today it is the most comprehensive and active reference database in pure and applied mathematics worldwide. Many prominent mathematicians have been... more... This book describes adaptive methods of statistical numerical analysis using evaluation of integrals, solution of integral equations, boundary value problems of the theory of elasticity and heat conduction as examples.The results and approaches provided in this book are different from those available in the literature as detailed descriptions of the... more... This is a book about the meaning of time, what it is, when it has started, how it flows and where to. It examines the consequences of Einstein's theory of relativity and offers startling suggestions about what recent research may reveal. more... The theory of linear Volterra Integro-differental equations has been developing rapidly in the last three decades. This book provides an easy-to-read, concise introduction to the theory of ill-posed abstract Volterra Integro-differential equations. It is accessible to readers whose backgrounds include functions of one complex variable, integration... more...
This eBook reviews some advanced topics in algebra, including exploring the nature of polynomials, functions, equations and identity's, examining the mathematical nomenclature used in multiplication and division. We consider multiplying out brackets, taking out common factors, manipulating algebraic fractions and simplifying expressions. We include an extensive selection of questions 4th grade students unique collection of Tunisian proverbs: Enjoyable, instructive, fun-to-share proverbs. Highly recommended to everyone, especially word and semantics lovers. After research, Carefully selected proverbs, translated as faithfully as possible, The author has deliberately chosen not to explain the proverbs, inviting the reader to use his/her own imagination and powerful interpretative skills! Short Essays on British Literature seeks to provide high school students, undergraduates, and general readers with a number of starting points from which to begin their own considerations of British literature, using clear, concise language and a minimum of scholarly jargon. Includes a helpful guide to citing each essay in MLA, Chicago, and APA style. This book is exclusively designed for Kids. The book consists of several questions,puzzles,tricky questions exclusively made for Kids. The book both entertains children and also provides knowledge on various topics.The book is real fun for Kids. Knowledge and fun provided in one single book for kids. The book is designed highly keeping in mind about kids and their way of learning. This book is for preachers, teachers, youth workers, leaders of holiday and after school programs and all others who are biblical storytellers. Action and emotion are key elements in capturing and maintain interest. This book focuses on the emotional interaction of the people involved in the events depicted in the Gospels. "As a man Mentoreth so is he" is a paraphrase of a well used aphorism. That's a great way to begin describing this volume on Mentoring. Mentorology Volume One is a textbook about Mentoring written like a storybook. Real life experiences in some extreme situations provide engaging and informative reading. As a Mentor, Teacher, Youth Leader or Coach you will learn from great Mentoring stories. This eBook introduces the subject of circle and circle geometry, introduces the equation of a circle, explores circle geometry, examines tangential lines to circles and their properties and equations, as well as exploring arc-length and sector area of circles where angles are represented in radians. Further, we include some elementary questions for the student to enjoy. Are you someone who "thinks things over?" Do you have conversations with yourself, silently in your head? If so, you are a practitioner of reflexivity. I use "classic" American films to describe four types of reflexivity. After introducing thirty-nine persons from the South Denver High School class of 1968, I ask if their type of reflexivity is related to the level of social status they reached. Practice and hone important subtraction skills with this firstWhat needs to be done to make America number one in education and productivity again? Why are our children lacking in these areas? Who is responsible? What does it mean to be responsible anymore? This book discusses the pitfalls of parents, teaches, administrators, and students in our country. Written by Janine Hunter, a former high school teacher or 32 years and Therapist of 3 years purpose of this book is to share with the reader a practical approach to and methodology for unbundling corporate management that will assist business leaders in improving their company's results and provide corporate executives with an employee empowerment management model.
Notes † - Items can only be purchased with PO or school credit card. Product Descriptions The Student Edition includes the same content as the on-level program, but is written at a reduced readability level with additional lesson and chapter support to ensure that struggling learners acquire key mathematics concepts. Student worktext contains the Student Companion and Practice and Problem Solving Workbook for the Foundations Series, all in one place plus one 6-year access to PowerAlgebra.com for a seemless transtion between digital and print. One 6-year access to PowerAlgebra.com serves as the portal into the digital world of Prentice Hall Algebra 1 Foundations Series. The online program can be used as part of a blended model with print, or as a digital course. Online access includes a wealth of assets, such as the Student Edition with audio, complete Teacher's Edition, instruction and presentation tools, editable worksheets, interactive online activities, student-generated videos, classroom management tools, and online assessments with remediation. This comprehensive resource contains the Student Companion and Practice and Problem Solving Workbook for the Foundations Series, all in one place. Provides complete daily support for every lesson. Includes Think About a Plan, Practice, and Standardized Test Prep. The 2-volumne printed set features teaching strategies offered in each lesson, supported by Grant Wiggins' Understanding by Design framework, additional instructional support for English Language Learners, struggling learners, and advanced learners, Chapter Planning Guides with activities and program resources labeled with levels of difficulty plus a Teacher's Resource DVD. Provides in-class support for new vocabulary, key concepts, Got Its, and lesson checks plus complete daily support for every lesson including Think About a Plan, practice, and standardized test preparation. Provides key features for each lesson and answers to Think About a Plan, Practice, and Standardized Test Prep.
23 --> 00:00:25 So let's start right away with stuff that we will need to see 8 00:00:25 --> 00:00:28 before we can go on to more advanced things. 9 00:00:28 --> 00:00:31 So, hopefully yesterday in recitation, you heard a bit 10 00:00:31 --> 00:00:34 about vectors. How many of you actually knew 11 00:00:34 --> 00:00:39 about vectors before that? OK, that's the vast majority. 12 00:00:39 --> 00:00:42 If you are not one of those people, well, 13 00:00:42 --> 00:00:45 hopefully you'll learn about vectors right now. 14 00:00:45 --> 00:00:48 I'm sorry that the learning curve will be a bit steeper for 15 00:00:48 --> 00:00:50 the first week. But hopefully, 16 00:00:50 --> 00:00:55 you'll adjust fine. If you have trouble with 17 00:00:55 --> 00:00:59 vectors, do go to your recitation instructor's office 18 00:00:59 --> 00:01:03 hours for extra practice if you feel the need to. 19 00:01:03 --> 00:01:09 You will see it's pretty easy. So, just to remind you, 20 00:01:09 --> 00:01:18 a vector is a quantity that has both a direction and a magnitude 21 00:01:18 --> 00:01:20 of length. 22 00:01:20 --> 00:01:33 23 00:01:33 --> 00:01:38 So -- So, concretely the way you draw a vector is by some 24 00:01:38 --> 00:01:40 arrow, like that, OK? 25 00:01:40 --> 00:01:43 And so, it has a length, and it's pointing in some 26 00:01:43 --> 00:01:45 direction. And, so, now, 27 00:01:45 --> 00:01:49 the way that we compute things with vectors, 28 00:01:49 --> 00:01:53 typically, as we introduce a coordinate system. 29 00:01:53 --> 00:01:57 So, if we are in the plane, x-y-axis, if we are in space, 30 00:01:57 --> 00:02:00 x-y-z axis. So, usually I will try to draw 31 00:02:00 --> 00:02:04 my x-y-z axis consistently to look like this. 32 00:02:04 --> 00:02:07 And then, I can represent my vector in terms of its 33 00:02:07 --> 00:02:10 components along the coordinate axis. 34 00:02:10 --> 00:02:13 So, that means when I have this row, I can ask, 35 00:02:13 --> 00:02:15 how much does it go in the x direction? 36 00:02:15 --> 00:02:17 How much does it go in the y direction? 37 00:02:17 --> 00:02:20 How much does it go in the z direction? 38 00:02:20 --> 00:02:25 And, so, let's call this a vector A. 39 00:02:25 --> 00:02:29 So, it's more convention. When we have a vector quantity, 40 00:02:29 --> 00:02:32 we put an arrow on top to remind us that it's a vector. 41 00:02:32 --> 00:02:35 If it's in the textbook, then sometimes it's in bold 42 00:02:35 --> 00:02:39 because it's easier to typeset. If you've tried in your 43 00:02:39 --> 00:02:44 favorite word processor, bold is easy and vectors are 44 00:02:44 --> 00:02:49 not easy. So, the vector you can try to 45 00:02:49 --> 00:02:56 decompose terms of unit vectors directed along the coordinate 46 00:02:56 --> 00:02:59 axis. So, the convention is there is 47 00:02:59 --> 00:03:03 a vector that we call *amplt;iampgt; 48 00:03:03 --> 00:03:08 hat that points along the x axis and has length one. 49 00:03:08 --> 00:03:10 There's a vector called amplt;jampgt; 50 00:03:10 --> 00:03:12 hat that does the same along the y axis, 51 00:03:12 --> 00:03:14 and the amplt;kampgt; 52 00:03:14 --> 00:03:16 hat that does the same along the z axis. 53 00:03:16 --> 00:03:20 And, so, we can express any vector in terms of its 54 00:03:20 --> 00:03:24 components. So, the other notation is 55 00:03:24 --> 00:03:29 amplt;a1, a2, a3 ampgt; 56 00:03:29 --> 00:03:37 between these square brackets. Well, in angular brackets. 57 00:03:37 --> 00:03:42 So, the length of a vector we denote by, if you want, 58 00:03:42 --> 00:03:47 it's the same notation as the absolute value. 59 00:03:47 --> 00:03:50 So, that's going to be a number, as we say, 60 00:03:50 --> 00:03:54 now, a scalar quantity. OK, so, a scalar quantity is a 61 00:03:54 --> 00:03:58 usual numerical quantity as opposed to a vector quantity. 62 00:03:58 --> 00:04:08 And, its direction is sometimes called dir A, 63 00:04:08 --> 00:04:13 and that can be obtained just by scaling the vector down to 64 00:04:13 --> 00:04:17 unit length, for example, 65 00:04:17 --> 00:04:26 by dividing it by its length. So -- Well, there's a lot of 66 00:04:26 --> 00:04:32 notation to be learned. So, for example, 67 00:04:32 --> 00:04:37 if I have two points, P and Q, then I can draw a 68 00:04:37 --> 00:04:42 vector from P to Q. And, that vector is called 69 00:04:42 --> 00:04:46 vector PQ, OK? So, maybe we'll call it A. 70 00:04:46 --> 00:04:48 But, a vector doesn't really have, necessarily, 71 00:04:48 --> 00:04:50 a starting point and an ending point. 72 00:04:50 --> 00:04:54 OK, so if I decide to start here and I go by the same 73 00:04:54 --> 00:04:57 distance in the same direction, this is also vector A. 74 00:04:57 --> 00:05:04 It's the same thing. So, a lot of vectors we'll draw 75 00:05:04 --> 00:05:08 starting at the origin, but we don't have to. 76 00:05:08 --> 00:05:19 So, let's just check and see how things went in recitation. 77 00:05:19 --> 00:05:23 So, let's say that I give you the vector 78 00:05:23 --> 00:05:34 amplt;3,2,1ampgt;. And so, what do you think about 79 00:05:34 --> 00:05:46 the length of this vector? OK, I see an answer forming. 80 00:05:46 --> 00:05:49 So, a lot of you are answering the same thing. 81 00:05:49 --> 00:05:54 Maybe it shouldn't spoil it for those who haven't given it yet. 82 00:05:54 --> 00:05:59 OK, I think the overwhelming vote is in favor of answer 83 00:05:59 --> 00:06:02 number two. I see some sixes, I don't know. 84 00:06:02 --> 00:06:06 That's a perfectly good answer, too, but hopefully in a few 85 00:06:06 --> 00:06:10 minutes it won't be I don't know anymore. 86 00:06:10 --> 00:06:17 So, let's see. How do we find -- -- the length 87 00:06:17 --> 00:06:24 of a vector three, two, one? 88 00:06:24 --> 00:06:30 Well, so, this vector, A, it comes towards us along 89 00:06:30 --> 00:06:37 the x axis by three units. It goes to the right along the 90 00:06:37 --> 00:06:42 y axis by two units, and then it goes up by one unit 91 00:06:42 --> 00:06:46 along the z axis. OK, so, it's pointing towards 92 00:06:46 --> 00:06:51 here. That's pretty hard to draw. 93 00:06:51 --> 00:06:55 So, how do we get its length? Well, maybe we can start with 94 00:06:55 --> 00:06:58 something easier, the length of the vector in the 95 00:06:58 --> 00:07:01 plane. So, observe that A is obtained 96 00:07:01 --> 00:07:04 from a vector, B, in the plane. 97 00:07:04 --> 00:07:09 Say, B equals three (i) hat plus two (j) hat. 98 00:07:09 --> 00:07:15 And then, we just have to, still, go up by one unit, 99 00:07:15 --> 00:07:17 OK? So, let me try to draw a 100 00:07:17 --> 00:07:20 picture in this vertical plane that contains A and B. 101 00:07:20 --> 00:07:23 If I draw it in the vertical plane, 102 00:07:23 --> 00:07:27 so, that's the Z axis, that's not any particular axis, 103 00:07:27 --> 00:07:38 then my vector B will go here, and my vector A will go above 104 00:07:38 --> 00:07:43 it. And here, that's one unit. 105 00:07:43 --> 00:07:49 And, here I have a right angle. So, I can use the Pythagorean 106 00:07:49 --> 00:07:57 theorem to find that length A^2 equals length B^2 plus one. 107 00:07:57 --> 00:08:00 Now, we are reduced to finding the length of B. 108 00:08:00 --> 00:08:02 The length of B, we can again find using the 109 00:08:02 --> 00:08:06 Pythagorean theorem in the XY plane because here we have the 110 00:08:06 --> 00:08:09 right angle. Here we have three units, 111 00:08:09 --> 00:08:12 and here we have two units. OK, so, if you do the 112 00:08:12 --> 00:08:15 calculations, you will see that, 113 00:08:15 --> 00:08:18 well, length of B is square root of (3^2 2^2), 114 00:08:18 --> 00:08:23 that's 13. So, the square root of 13 -- -- 115 00:08:23 --> 00:08:32 and length of A is square root of length B^2 plus one (square 116 00:08:32 --> 00:08:41 it if you want) which is going to be square root of 13 plus one 117 00:08:41 --> 00:08:49 is the square root of 14, hence, answer number two which 118 00:08:49 --> 00:08:54 almost all of you gave. OK, so the general formula, 119 00:08:54 --> 00:09:02 if you follow it with it, in general if we have a vector 120 00:09:02 --> 00:09:07 with components a1, a2, a3, 121 00:09:07 --> 00:09:16 then the length of A is the square root of a1^2 plus a2^2 122 00:09:16 --> 00:09:23 plus a3^2. OK, any questions about that? 123 00:09:23 --> 00:09:29 Yes? Yes. 124 00:09:29 --> 00:09:32 So, in general, we indeed can consider vectors 125 00:09:32 --> 00:09:36 in abstract spaces that have any number of coordinates. 126 00:09:36 --> 00:09:38 And that you have more components. 127 00:09:38 --> 00:09:40 In this class, we'll mostly see vectors with 128 00:09:40 --> 00:09:44 two or three components because they are easier to draw, 129 00:09:44 --> 00:09:47 and because a lot of the math that we'll see works exactly the 130 00:09:47 --> 00:09:50 same way whether you have three variables or a million 131 00:09:50 --> 00:09:52 variables. If we had a factor with more 132 00:09:52 --> 00:09:55 components, then we would have a lot of trouble drawing it. 133 00:09:55 --> 00:09:58 But we could still define its length in the same way, 134 00:09:58 --> 00:10:01 by summing the squares of the components. 135 00:10:01 --> 00:10:04 So, I'm sorry to say that here, multi-variable, 136 00:10:04 --> 00:10:07 multi will mean mostly two or three. 137 00:10:07 --> 00:10:13 But, be assured that it works just the same way if you have 138 00:10:13 --> 00:10:20 10,000 variables. Just, calculations are longer. 139 00:10:20 --> 00:10:28 OK, more questions? So, what else can we do with 140 00:10:28 --> 00:10:31 vectors? Well, another thing that I'm 141 00:10:31 --> 00:10:35 sure you know how to do with vectors is to add them to scale 142 00:10:35 --> 00:10:39 them. So, vector addition, 143 00:10:39 --> 00:10:48 so, if you have two vectors, A and B, then you can form, 144 00:10:48 --> 00:10:52 their sum, A plus B. How do we do that? 145 00:10:52 --> 00:10:54 Well, first, I should tell you, 146 00:10:54 --> 00:10:56 vectors, they have this double life. 147 00:10:56 --> 00:10:59 They are, at the same time, geometric objects that we can 148 00:10:59 --> 00:11:02 draw like this in pictures, and there are also 149 00:11:02 --> 00:11:06 computational objects that we can represent by numbers. 150 00:11:06 --> 00:11:09 So, every question about vectors will have two answers, 151 00:11:09 --> 00:11:11 one geometric, and one numerical. 152 00:11:11 --> 00:11:14 OK, so let's start with the geometric. 153 00:11:14 --> 00:11:17 So, let's say that I have two vectors, A and B, 154 00:11:17 --> 00:11:21 given to me. And, let's say that I thought 155 00:11:21 --> 00:11:24 of drawing them at the same place to start with. 156 00:11:24 --> 00:11:28 Well, to take the sum, what I should do is actually 157 00:11:28 --> 00:11:33 move B so that it starts at the end of A, at the head of A. 158 00:11:33 --> 00:11:38 OK, so this is, again, vector B. So, observe, 159 00:11:38 --> 00:11:41 this actually forms, now, a parallelogram, 160 00:11:41 --> 00:11:43 right? So, this side is, 161 00:11:43 --> 00:11:48 again, vector A. And now, if we take the 162 00:11:48 --> 00:11:57 diagonal of that parallelogram, this is what we call A plus B, 163 00:11:57 --> 00:12:00 OK, so, the idea being that to move along A plus B, 164 00:12:00 --> 00:12:03 it's the same as to move first along A and then along B, 165 00:12:03 --> 00:12:09 or, along B, then along A. A plus B equals B plus A. 166 00:12:09 --> 00:12:13 OK, now, if we do it numerically, 167 00:12:13 --> 00:12:19 then all you do is you just add the first component of A with 168 00:12:19 --> 00:12:23 the first component of B, the second with the second, 169 00:12:23 --> 00:12:28 and the third with the third. OK, say that A was 170 00:12:28 --> 00:12:31 amplt;a1, a2, a3ampgt; 171 00:12:31 --> 00:12:35 B was amplt;b1, b2, b3ampgt;, 172 00:12:35 --> 00:12:40 then you just add this way. OK, so it's pretty 173 00:12:40 --> 00:12:44 straightforward. So, for example, 174 00:12:44 --> 00:12:48 I said that my vector over there, its components are three, 175 00:12:48 --> 00:12:54 two, one. But, I also wrote it as 3i 2j k. 176 00:12:54 --> 00:12:57 What does that mean? OK, so I need to tell you first 177 00:12:57 --> 00:13:06 about multiplying by a scalar. So, this is about addition. 178 00:13:06 --> 00:13:11 So, multiplication by a scalar, it's very easy. 179 00:13:11 --> 00:13:15 If you have a vector, A, then you can form a vector 180 00:13:15 --> 00:13:20 2A just by making it go twice as far in the same direction. 181 00:13:20 --> 00:13:24 Or, we can make half A more modestly. 182 00:13:24 --> 00:13:31 We can even make minus A, and so on. 183 00:13:31 --> 00:13:35 So now, you see, if I do the calculation, 184 00:13:35 --> 00:13:38 3i 2j k, well, what does it mean? 185 00:13:38 --> 00:13:43 3i is just going to go along the x axis, but by distance of 186 00:13:43 --> 00:13:47 three instead of one. And then, 2j goes two units 187 00:13:47 --> 00:13:51 along the y axis, and k goes up by one unit. 188 00:13:51 --> 00:13:54 Well, if you add these together, you will go from the 189 00:13:54 --> 00:13:58 origin, then along the x axis, then parallel to the y axis, 190 00:13:58 --> 00:14:02 and then up. And, you will end up, 191 00:14:02 --> 00:14:05 indeed, at the endpoint of a vector. 192 00:14:05 --> 00:14:19 OK, any questions at this point? Yes? 193 00:14:19 --> 00:14:21 Exactly. To add vectors geometrically, 194 00:14:21 --> 00:14:25 you just put the head of the first vector and the tail of the 195 00:14:25 --> 00:14:30 second vector in the same place. And then, it's head to tail 196 00:14:30 --> 00:14:35 addition. Any other questions? 197 00:14:35 --> 00:14:41 Yes? That's correct. 198 00:14:41 --> 00:14:43 If you subtract two vectors, that just means you add the 199 00:14:43 --> 00:14:45 opposite of a vector. So, for example, 200 00:14:45 --> 00:14:49 if I wanted to do A minus B, I would first go along A and 201 00:14:49 --> 00:14:52 then along minus B, which would take me somewhere 202 00:14:52 --> 00:14:55 over there, OK? So, A minus B, 203 00:14:55 --> 00:15:01 if you want, would go from here to here. 204 00:15:01 --> 00:15:08 OK, so hopefully you've kind of seen that stuff either before in 205 00:15:08 --> 00:15:13 your lives, or at least yesterday. 206 00:15:13 --> 00:15:23 So, I'm going to use that as an excuse to move quickly forward. 207 00:15:23 --> 00:15:28 So, now we are going to learn a few more operations about 208 00:15:28 --> 00:15:31 vectors. And, these operations will be 209 00:15:31 --> 00:15:34 useful to us when we start trying to do a bit of geometry. 210 00:15:34 --> 00:15:37 So, of course, you've all done some geometry. 211 00:15:37 --> 00:15:40 But, we are going to see that geometry can be done using 212 00:15:40 --> 00:15:42 vectors. And, in many ways, 213 00:15:42 --> 00:15:44 it's the right language for that, 214 00:15:44 --> 00:15:47 and in particular when we learn about functions we really will 215 00:15:47 --> 00:15:51 want to use vectors more than, maybe, the other kind of 216 00:15:51 --> 00:15:54 geometry that you've seen before. 217 00:15:54 --> 00:15:56 I mean, of course, it's just a language in a way. 218 00:15:56 --> 00:15:59 I mean, we are just reformulating things that you 219 00:15:59 --> 00:16:02 have seen, you already know since childhood. 220 00:16:02 --> 00:16:07 But, you will see that notation somehow helps to make it more 221 00:16:07 --> 00:16:10 straightforward. So, what is dot product? 222 00:16:10 --> 00:16:16 Well, dot product as a way of multiplying two vectors to get a 223 00:16:16 --> 00:16:21 number, a scalar. And, well, let me start by 224 00:16:21 --> 00:16:25 giving you a definition in terms of components. 225 00:16:25 --> 00:16:29 What we do, let's say that we have a vector, 226 00:16:29 --> 00:16:32 A, with components a1, a2, a3, vector B with 227 00:16:32 --> 00:16:34 components b1, b2, b3. 228 00:16:34 --> 00:16:38 Well, we multiply the first components by the first 229 00:16:38 --> 00:16:43 components, the second by the second, the third by the third. 230 00:16:43 --> 00:16:46 If you have N components, you keep going. 231 00:16:46 --> 00:16:49 And, you sum all of these together. 232 00:16:49 --> 00:16:55 OK, and important: this is a scalar. 233 00:16:55 --> 00:16:59 OK, you do not get a vector. You get a number. 234 00:16:59 --> 00:17:01 I know it sounds completely obvious from the definition 235 00:17:01 --> 00:17:03 here, but in the middle of the action 236 00:17:03 --> 00:17:07 when you're going to do complicated problems, 237 00:17:07 --> 00:17:14 it's sometimes easy to forget. So, that's the definition. 238 00:17:14 --> 00:17:17 What is it good for? Why would we ever want to do 239 00:17:17 --> 00:17:20 that? That's kind of a strange 240 00:17:20 --> 00:17:23 operation. So, probably to see what it's 241 00:17:23 --> 00:17:27 good for, I should first tell you what it is geometrically. 242 00:17:27 --> 00:17:29 OK, so what does it do geometrically? 243 00:17:29 --> 00:17:38 244 00:17:38 --> 00:17:42 Well, what you do when you multiply two vectors in this 245 00:17:42 --> 00:17:45 way, I claim the answer is equal to 246 00:17:45 --> 00:17:51 the length of A times the length of B times the cosine of the 247 00:17:51 --> 00:17:59 angle between them. So, I have my vector, A, 248 00:17:59 --> 00:18:04 and if I have my vector, B, and I have some angle between 249 00:18:04 --> 00:18:06 them, I multiply the length of A 250 00:18:06 --> 00:18:10 times the length of B times the cosine of that angle. 251 00:18:10 --> 00:18:13 So, that looks like a very artificial operation. 252 00:18:13 --> 00:18:16 I mean, why would want to do that complicated multiplication? 253 00:18:16 --> 00:18:21 Well, the basic answer is it tells us at the same time about 254 00:18:21 --> 00:18:25 lengths and about angles. And, the extra bonus thing is 255 00:18:25 --> 00:18:29 that it's very easy to compute if you have components, 256 00:18:29 --> 00:18:32 see, that formula is actually pretty easy. 257 00:18:32 --> 00:18:39 So, OK, maybe I should first tell you, how do we get this 258 00:18:39 --> 00:18:41 from that? Because, you know, 259 00:18:41 --> 00:18:44 in math, one tries to justify everything to prove theorems. 260 00:18:44 --> 00:18:45 So, if you want, that's the theorem. 261 00:18:45 --> 00:18:47 That's the first theorem in 18.02. 262 00:18:47 --> 00:18:52 So, how do we prove the theorem? How do we check that this is, 263 00:18:52 --> 00:18:55 indeed, correct using this definition? 264 00:18:55 --> 00:19:06 So, in more common language, what does this geometric 265 00:19:06 --> 00:19:11 definition mean? Well, the first thing it means, 266 00:19:11 --> 00:19:14 before we multiply two vectors, let's start multiplying a 267 00:19:14 --> 00:19:17 vector with itself. That's probably easier. 268 00:19:17 --> 00:19:19 So, if we multiply a vector, A, with itself, 269 00:19:19 --> 00:19:22 using this dot product, so, by the way, 270 00:19:22 --> 00:19:24 I should point out, we put this dot here. 271 00:19:24 --> 00:19:28 That's why it's called dot product. 272 00:19:28 --> 00:19:33 So, what this tells us is we should get the same thing as 273 00:19:33 --> 00:19:38 multiplying the length of A with itself, so, squared, 274 00:19:38 --> 00:19:43 times the cosine of the angle. But now, the cosine of an 275 00:19:43 --> 00:19:49 angle, of zero, cosine of zero you all know is 276 00:19:49 --> 00:19:52 one. OK, so that's going to be 277 00:19:52 --> 00:19:56 length A^2. Well, doesn't stand a chance of 278 00:19:56 --> 00:19:57 being true? Well, let's see. 279 00:19:57 --> 00:20:03 If we do AdotA using this formula, we will get a1^2 a2^2 280 00:20:03 --> 00:20:07 a3^2. That is, indeed, 281 00:20:07 --> 00:20:14 the square of the length. So, check. 282 00:20:14 --> 00:20:18 That works. OK, now, what about two 283 00:20:18 --> 00:20:23 different vectors? Can we understand what this 284 00:20:23 --> 00:20:27 says, and how it relates to that? 285 00:20:27 --> 00:20:33 So, let's say that I have two different vectors, 286 00:20:33 --> 00:20:40 A and B, and I want to try to understand what's going on. 287 00:20:40 --> 00:20:45 So, my claim is that we are going to be able to understand 288 00:20:45 --> 00:20:49 the relation between this and that in terms of the law of 289 00:20:49 --> 00:20:52 cosines. So, the law of cosines is 290 00:20:52 --> 00:20:56 something that tells you about the length of the third side in 291 00:20:56 --> 00:21:00 the triangle like this in terms of these two sides, 292 00:21:00 --> 00:21:07 and the angle here. OK, so the law of cosines, 293 00:21:07 --> 00:21:11 which hopefully you have seen before, says that, 294 00:21:11 --> 00:21:14 so let me give a name to this side. 295 00:21:14 --> 00:21:19 Let's call this side C, and as a vector, 296 00:21:19 --> 00:21:29 C is A minus B. It's minus B plus A. 297 00:21:29 --> 00:21:37 So, it's getting a bit cluttered here. 298 00:21:37 --> 00:21:45 So, the law of cosines says that the length of the third 299 00:21:45 --> 00:21:53 side in this triangle is equal to length A2 plus length B2. 300 00:21:53 --> 00:21:56 Well, if I stopped here, that would be Pythagoras, 301 00:21:56 --> 00:22:01 but I don't have a right angle. So, I have a third term which 302 00:22:01 --> 00:22:07 is twice length A, length B, cosine theta, 303 00:22:07 --> 00:22:10 OK? Has everyone seen this formula 304 00:22:10 --> 00:22:13 sometime? I hear some yeah's. 305 00:22:13 --> 00:22:16 I hear some no's. Well, it's a fact about, 306 00:22:16 --> 00:22:19 I mean, you probably haven't seen it with vectors, 307 00:22:19 --> 00:22:22 but it's a fact about the side lengths in a triangle. 308 00:22:22 --> 00:22:27 And, well, let's say, if you haven't seen it before, 309 00:22:27 --> 00:22:32 then this is going to be a proof of the law of cosines if 310 00:22:32 --> 00:22:39 you believe this. Otherwise, it's the other way 311 00:22:39 --> 00:22:43 around. So, let's try to see how this 312 00:22:43 --> 00:22:47 relates to what I'm saying about the dot product. 313 00:22:47 --> 00:22:54 So, I've been saying that length C^2, that's the same 314 00:22:54 --> 00:22:56 thing as CdotC, OK? 315 00:22:56 --> 00:23:01 That, we have checked. Now, CdotC, well, 316 00:23:01 --> 00:23:06 C is A minus B. So, it's A minus B, 317 00:23:06 --> 00:23:09 dot product, A minus B. 318 00:23:09 --> 00:23:11 Now, what do we want to do in a situation like that? 319 00:23:11 --> 00:23:16 Well, we want to expand this into a sum of four terms. 320 00:23:16 --> 00:23:19 Are we allowed to do that? Well, we have this dot product 321 00:23:19 --> 00:23:22 that's a mysterious new operation. 322 00:23:22 --> 00:23:24 We don't really know. Well, the answer is yes, 323 00:23:24 --> 00:23:27 we can do it. You can check from this 324 00:23:27 --> 00:23:31 definition that it behaves in the usual way in terms of 325 00:23:31 --> 00:23:34 expanding, vectoring, and so on. 326 00:23:34 --> 00:23:49 So, I can write that as AdotA minus AdotB minus BdotA plus 327 00:23:49 --> 00:23:55 BdotB. So, AdotA is length A^2. 328 00:23:55 --> 00:23:56 Let me jump ahead to the last term. 329 00:23:56 --> 00:24:01 BdotB is length B^2, and then these two terms, 330 00:24:01 --> 00:24:04 well, they're the same. You can check from the 331 00:24:04 --> 00:24:07 definition that AdotB and BdotA are the same thing. 332 00:24:07 --> 00:24:20 333 00:24:20 --> 00:24:24 Well, you see that this term, I mean, this is the only 334 00:24:24 --> 00:24:30 difference between these two formulas for the length of C. 335 00:24:30 --> 00:24:34 So, if you believe in the law of cosines, then it tells you 336 00:24:34 --> 00:24:39 that, yes, this a proof that AdotB equals length A length B 337 00:24:39 --> 00:24:41 cosine theta. Or, vice versa, 338 00:24:41 --> 00:24:45 if you've never seen the law of cosines, you are willing to 339 00:24:45 --> 00:24:49 believe this. Then, this is the proof of the 340 00:24:49 --> 00:24:53 law of cosines. So, the law of cosines, 341 00:24:53 --> 00:24:59 or this interpretation, are equivalent to each other. 342 00:24:59 --> 00:25:07 OK, any questions? Yes? 343 00:25:07 --> 00:25:12 So, in the second one there isn't a cosine theta because I'm 344 00:25:12 --> 00:25:16 just expanding a dot product. OK, so I'm just writing C 345 00:25:16 --> 00:25:19 equals A minus B, and then I'm expanding this 346 00:25:19 --> 00:25:22 algebraically. And then, I get to an answer 347 00:25:22 --> 00:25:24 that has an A.B. So then, if I wanted to express 348 00:25:24 --> 00:25:27 that without a dot product, then I would have to introduce 349 00:25:27 --> 00:25:31 a cosine. And, I would get the same as 350 00:25:31 --> 00:25:34 that, OK? So, yeah, if you want, 351 00:25:34 --> 00:25:38 the next step to recall the law of cosines would be plug in this 352 00:25:38 --> 00:25:43 formula for AdotB. And then you would have a 353 00:25:43 --> 00:25:58 cosine. OK, let's keep going. 354 00:25:58 --> 00:26:03 OK, so what is this good for? Now that we have a definition, 355 00:26:03 --> 00:26:06 we should figure out what we can do with it. 356 00:26:06 --> 00:26:11 So, what are the applications of dot product? 357 00:26:11 --> 00:26:14 Well, will this discover new applications of dot product 358 00:26:14 --> 00:26:17 throughout the entire semester,but let me tell you at 359 00:26:17 --> 00:26:20 least about those that are readily visible. 360 00:26:20 --> 00:26:33 So, one is to compute lengths and angles, especially angles. 361 00:26:33 --> 00:26:39 So, let's do an example. Let's say that, 362 00:26:39 --> 00:26:44 for example, I have in space, 363 00:26:44 --> 00:26:51 I have a point, P, which is at (1,0,0). 364 00:26:51 --> 00:26:55 I have a point, Q, which is at (0,1,0). 365 00:26:55 --> 00:26:58 So, it's at distance one here, one here. 366 00:26:58 --> 00:27:03 And, I have a third point, R at (0,0,2), 367 00:27:03 --> 00:27:07 so it's at height two. And, let's say that I'm 368 00:27:07 --> 00:27:11 curious, and I'm wondering what is the angle here? 369 00:27:11 --> 00:27:15 So, here I have a triangle in space connect P, 370 00:27:15 --> 00:27:20 Q, and R, and I'm wondering, what is this angle here? 371 00:27:20 --> 00:27:23 OK, so, of course, one solution is to build a 372 00:27:23 --> 00:27:25 model and then go and measure the angle. 373 00:27:25 --> 00:27:28 But, we can do better than that. We can just find the angle 374 00:27:28 --> 00:27:32 using dot product. So, how would we do that? 375 00:27:32 --> 00:27:38 Well, so, if we look at this formula, we see, 376 00:27:38 --> 00:27:44 so, let's say that we want to find the angle here. 377 00:27:44 --> 00:27:50 Well, let's look at the formula for PQdotPR. 378 00:27:50 --> 00:27:56 Well, we said it should be length PQ times length PR times 379 00:27:56 --> 00:27:59 the cosine of the angle, OK? 380 00:27:59 --> 00:28:01 Now, what do we know, and what do we not know? 381 00:28:01 --> 00:28:04 Well, certainly at this point we don't know the cosine of the 382 00:28:04 --> 00:28:06 angle. That's what we would like to 383 00:28:06 --> 00:28:08 find. The lengths, 384 00:28:08 --> 00:28:11 certainly we can compute. We know how to find these 385 00:28:11 --> 00:28:14 lengths. And, this dot product we know 386 00:28:14 --> 00:28:17 how to compute because we have an easy formula here. 387 00:28:17 --> 00:28:20 OK, so we can compute everything else and then find 388 00:28:20 --> 00:28:25 theta. So, I'll tell you what we will 389 00:28:25 --> 00:28:31 do is we will find theta -- -- in this way. 390 00:28:31 --> 00:28:34 We'll take the dot product of PQ with PR, and then we'll 391 00:28:34 --> 00:28:36 divide by the lengths. 392 00:28:36 --> 00:29:14 393 00:29:14 --> 00:29:27 OK, so let's see. So, we said cosine theta is 394 00:29:27 --> 00:29:33 PQdotPR over length PQ length PR. 395 00:29:33 --> 00:29:36 So, let's try to figure out what this vector, 396 00:29:36 --> 00:29:39 PQ, well, to go from P to Q, 397 00:29:39 --> 00:29:43 I should go minus one unit along the x direction plus one 398 00:29:43 --> 00:29:46 unit along the y direction. And, I'm not moving in the z 399 00:29:46 --> 00:29:49 direction. So, to go from P to Q, 400 00:29:49 --> 00:29:54 I have to move by amplt;-1,1,0amp*gt;. 401 00:29:54 --> 00:29:59 To go from P to R, I go -1 along the x axis and 2 402 00:29:59 --> 00:30:04 along the z axis. So, PR, I claim, is this. 403 00:30:04 --> 00:30:12 OK, then, the lengths of these vectors, well,(-1)^2 (1)^2 404 00:30:12 --> 00:30:19 (0)^2, square root, and then same thing with the 405 00:30:19 --> 00:30:24 other one. OK, so, the denominator will 406 00:30:24 --> 00:30:30 become the square root of 2, and there's a square root of 5. 407 00:30:30 --> 00:30:34 What about the numerator? Well, so, remember, 408 00:30:34 --> 00:30:37 to do the dot product, we multiply this by this, 409 00:30:37 --> 00:30:40 and that by that, that by that. 410 00:30:40 --> 00:30:45 And, we add. Minus 1 times minus 1 makes 1 411 00:30:45 --> 00:30:49 plus 1 times 0, that's 0. 412 00:30:49 --> 00:30:55 Zero times 2 is 0 again. So, we will get 1 over square 413 00:30:55 --> 00:30:59 root of 10. That's the cosine of the angle. 414 00:30:59 --> 00:31:03 And, of course if we want the actual angle, 415 00:31:03 --> 00:31:08 well, we have to take a calculator, find the inverse 416 00:31:08 --> 00:31:12 cosine, and you'll find it's about 71.5°. 417 00:31:12 --> 00:31:18 Actually, we'll be using mostly radians, but for today, 418 00:31:18 --> 00:31:26 that's certainly more speaking. OK, any questions about that? 419 00:31:26 --> 00:31:29 No? OK, so in particular, 420 00:31:29 --> 00:31:32 I should point out one thing that's really neat about the 421 00:31:32 --> 00:31:34 answer. I mean, we got this number. 422 00:31:34 --> 00:31:37 We don't really know what it means exactly because it mixes 423 00:31:37 --> 00:31:39 together the lengths and the angle. 424 00:31:39 --> 00:31:41 But, one thing that's interesting here, 425 00:31:41 --> 00:31:45 it's the sign of the answer, the fact that we got a positive 426 00:31:45 --> 00:31:48 number. So, if you think about it, 427 00:31:48 --> 00:31:50 the lengths are always positive. 428 00:31:50 --> 00:31:56 So, the sign of a dot product is the same as a sign of cosine 429 00:31:56 --> 00:32:00 theta. So, in fact, 430 00:32:00 --> 00:32:13 the sign of AdotB is going to be positive if the angle is less 431 00:32:13 --> 00:32:17 than 90°. So, that means geometrically, 432 00:32:17 --> 00:32:21 my two vectors are going more or less in the same direction. 433 00:32:21 --> 00:32:27 They make an acute angle. It's going to be zero if the 434 00:32:27 --> 00:32:33 angle is exactly 90°, OK, because that's when the 435 00:32:33 --> 00:32:39 cosine will be zero. And, it will be negative if the 436 00:32:39 --> 00:32:43 angle is more than 90°. So, that means they go, 437 00:32:43 --> 00:32:46 however, in opposite directions. 438 00:32:46 --> 00:32:50 So, that's basically one way to think about what dot product 439 00:32:50 --> 00:32:54 measures. It measures how much the two 440 00:32:54 --> 00:32:58 vectors are going along each other. 441 00:32:58 --> 00:33:02 OK, and that actually leads us to the next application. 442 00:33:02 --> 00:33:05 So, let's see, did I have a number one there? 443 00:33:05 --> 00:33:07 Yes. So, if I had a number one, 444 00:33:07 --> 00:33:12 I must have number two. The second application is to 445 00:33:12 --> 00:33:16 detect orthogonality. It's to figure out when two 446 00:33:16 --> 00:33:21 things are perpendicular. OK, so orthogonality is just a 447 00:33:21 --> 00:33:26 complicated word from Greek to say things are perpendicular. 448 00:33:26 --> 00:33:34 So, let's just take an example. Let's say I give you the 449 00:33:34 --> 00:33:41 equation x 2y 3z = 0. OK, so that defines a certain 450 00:33:41 --> 00:33:46 set of points in space, and what do you think the set 451 00:33:46 --> 00:33:52 of solutions look like if I give you this equation? 452 00:33:52 --> 00:34:01 So far I see one, two, three answers, 453 00:34:01 --> 00:34:06 OK. So, I see various competing 454 00:34:06 --> 00:34:11 answers, but, yeah, I see a lot of people 455 00:34:11 --> 00:34:18 voting for answer number four. I see also some I don't knows, 456 00:34:18 --> 00:34:22 and some other things. But, the majority vote seems to 457 00:34:22 --> 00:34:26 be a plane. And, indeed that's the correct 458 00:34:26 --> 00:34:28 answer. So, how do we see that it's a 459 00:34:28 --> 00:34:28 plane? 460 00:34:28 --> 00:34:43 461 00:34:43 --> 00:34:49 So, I should say, this is the equation of a 462 00:34:49 --> 00:34:52 plane. So, there's many ways to see 463 00:34:52 --> 00:34:55 that, and I'm not going to give you all of them. 464 00:34:55 --> 00:34:58 But, here's one way to think about it. 465 00:34:58 --> 00:35:03 So, let's think geometrically about how to express this 466 00:35:03 --> 00:35:09 condition in terms of vectors. So, let's take the origin O, 467 00:35:09 --> 00:35:13 by convention is the point (0,0,0). 468 00:35:13 --> 00:35:18 And, let's take a point, P, that will satisfy this 469 00:35:18 --> 00:35:21 equation on it, so, at coordinates x, 470 00:35:21 --> 00:35:24 y, z. So, what does this condition 471 00:35:24 --> 00:35:28 here mean? Well, it means the following 472 00:35:28 --> 00:35:32 thing. So, let's take the vector, OP. 473 00:35:32 --> 00:35:37 OK, so vector OP, of course, has components x, 474 00:35:37 --> 00:35:40 y, z. Now, we can think of this as 475 00:35:40 --> 00:35:44 actually a dot product between OP and a mysterious vector that 476 00:35:44 --> 00:35:47 won't remain mysterious for very long, 477 00:35:47 --> 00:35:50 namely, the vector one, two, three. 478 00:35:50 --> 00:35:59 OK, so, this condition is the same as OP.A equals zero, 479 00:35:59 --> 00:36:03 right? If I take the dot product 480 00:36:03 --> 00:36:09 OPdotA I get x times one plus y times two plus z times three. 481 00:36:09 --> 00:36:14 But now, what does it mean that the dot product between OP and A 482 00:36:14 --> 00:36:19 is zero? Well, it means that OP and A 483 00:36:19 --> 00:36:25 are perpendicular. OK, so I have this vector, A. 484 00:36:25 --> 00:36:28 I'm not going to be able to draw it realistically. 485 00:36:28 --> 00:36:32 Let's say it goes this way. Then, a point, 486 00:36:32 --> 00:36:37 P, solves this equation exactly when the vector from O to P is 487 00:36:37 --> 00:36:40 perpendicular to A. And, I claim that defines a 488 00:36:40 --> 00:36:41 plane. For example, 489 00:36:41 --> 00:36:45 if it helps you to see it, take a vertical vector. 490 00:36:45 --> 00:36:47 What does it mean to be perpendicular to the vertical 491 00:36:47 --> 00:36:49 vector? It means you are horizontal. 492 00:36:49 --> 00:36:56 It's the horizontal plane. Here, it's a plane that passes 493 00:36:56 --> 00:37:05 through the origin and is perpendicular to this vector, 494 00:37:05 --> 00:37:14 A. OK, so what we get is a plane 495 00:37:14 --> 00:37:25 through the origin perpendicular to A. 496 00:37:25 --> 00:37:29 And, in general, what you should remember is 497 00:37:29 --> 00:37:35 that two vectors have a dot product equal to zero if and 498 00:37:35 --> 00:37:41 only if that's equivalent to the cosine of the angle between them 499 00:37:41 --> 00:37:46 is zero. That means the angle is 90°. 500 00:37:46 --> 00:37:51 That means A and B are perpendicular. 501 00:37:51 --> 00:37:57 So, we have a very fast way of checking whether two vectors are 502 00:37:57 --> 00:38:01 perpendicular. So, one additional application 503 00:38:01 --> 00:38:05 I think we'll see actually tomorrow is to find the 504 00:38:05 --> 00:38:10 components of a vector along a certain direction. 505 00:38:10 --> 00:38:13 So, I claim we can use this intuition I gave about dot 506 00:38:13 --> 00:38:16 product telling us how much to vectors go in the same direction 507 00:38:16 --> 00:38:19 to actually give a precise meaning to the notion of 508 00:38:19 --> 00:38:22 component for vector, not just along the x, 509 00:38:22 --> 00:38:27 y, or z axis, but along any direction in 510 00:38:27 --> 00:38:31 space. So, I think I should probably 511 00:38:31 --> 00:38:34 stop here. But, I will see you tomorrow at 512 00:38:34 --> 00:38:38 2:00 here, and we'll learn more about that and about cross 513 00:38:38 --> 00:38:44 products. 514 00:38:44 --> 00:38:49
OpsResearch - DRA Systems A collection of Java classes for developing operations research programs and other mathematical applications. The site includes documentation and tutorials, and software download is free. Also features a bookstore and related links. ...more>> Pat Thompson's Home Page - Patrick W. Thompson Research on students' development of algebra and calculus ideas in grades 3-16, on the nature of multiplicative reasoning and its role in learning sophisticated mathematical ideas, and on teaching and learning statistical and probabilistic reasoning. ...more>> Picking Winners - Ivars Peterson (MathTrek) One of the attractive features of spectator sports is the uncertainty of the outcome. Even when one team is overwhelmingly favored to win, the underdog may still come through with a surprising victory. Nonetheless, the ability to pick winners can be of ...more>> Pitfalls of Data Analysis - Clay Helberg Subtitled "... or How to Avoid Lies and Damned Lies," this paper discusses what people "often overlook in their data analysis, and ways people sometimes 'bend the rules' of statistics to support their viewpoint." Tips to ensure the clarity and accuracy Pop Clock - Susan Boone A lesson on data collection, problem solving, research skills, interpolation of data. Students review the Census Bureau's home page on the Internet and gather data regarding trends in population. They study these data and make predictions of future populations, ...more>> The Power of One - Robert Matthews, The New Scientist Benford's law demands that around 30 per cent of the numbers in a given data set will start with a 1, 18 per cent with a 2, right down to just 4.6 per cent starting with a 9. This essay provides a mathematical history behind the counterintuitive law, ...more>> PreCalculus Problem of the Week - Math Forum Math problems for students who had finished studying topics commonly covered in first-year algebra and high school geometry. From 2002 to 2003, problems involved probability, statistics, discrete math, and trigonometry. The goal was to challenge students ...more>>
Synopses & Reviews Publisher Comments: This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis. The authors were well-known for their powers of exposition and made this subject accessible to a wide audience of mathematicians. Synopsis: "Synopsis" by Gardners,
Hello! I have been interested in physics for a while and have read many books and seen documentaries about physics in general. Unfortunately, I have neglected to get into the math and real details of what is really going on in these interesting concepts such as quantum mechanics and general relativity. I was wondering if anyone could recommend books or textbooks that could give me an introduction perhaps as to how these things work mathematically. Preferably nothing too advanced (I am in basic high school algebra and geometry, but can catch on to things quickly if necessary).Any advice or recommendations could help! Thank you very much. After a couple semesters in calculus, try on "Mathematical Methods in the Physical Sciences" by Mary Boas. It starts with some concepts that aren't too difficult but will prepare you for all of the special differential equations that relate to solving sophomore/junior level Physics courses. The foundational mathematics for physics is calculus, differential equations and linear algebra. If your doing hs algebra/geometry atm, your next step would be single variable calculus. MIT OCW has a great set of lectures in each of these subjects to help get started. Keep in mind though that you typically want to have most of the algebra and geometry down packed and understood before you do calc. The stuff that your learning now will come up again and again in all other mathematics. Also, the topics you listed (quantum mechanics, general relativity) require some fairly advanced mathematics relative you what your learning now so take it slow and get a solid understanding of the basics. There's nothing worse then half-assing your basics and then trying to build on a shitty foundation. You'll crumble when you reach quantum mechanics. PS give special relativity a try when you get some basic calc down. It's quite simple and has tons of cool applications. Leave the real math behind general relativity alone till you get a degree in math or physics :P A pretty well written math methods text (recently published by Dover but still available online for free). Covers many topics in, for example, Boas. It's written more "tutorial" style in my opinion than Boas. Depending on your learning style you might like it - and it's free. Give it a go! You'll need to get through calculus to really be able to understand the math that's going on in more advanced physics. In the mean time I would recommend studying basic physics at an algebraic level. If you want something more that that, then I would got to MIT's website and track down the lectures and notes that are posted there. It may be easier to to learn what you'd like to know by looking at those then compared to an advanced mathematical physics text. Have Riley, Hobson and Bence and used it a lot during 4th year undergraduate (I don't know what that corresponds to in US system, sorry) I still dip into it from time to time. I also quite like [Quantum Mechanics by Rae] But there is a lot of material on the internet available for free. I remember quite liking Pauls online math's notes for calculus learnin'. And, for example, there are Stanford's videos of modern physics lecture courses. I think MIT do this kind of thing too.
"Rocket scientists wanted" A comprehensive scientific calculator app Yes, your droid comes with a calculator and you might even already have an additional calculator app. However, if you are surviving high school or are in any scientific degree in college, those toy calculators aren't enough. You need to solve quadratic equations and polynomials and vectors and matrixes and draw graphs and all that odd Math stuff we mere mortals can't understand. Perhaps you need to save it all and bring it with you to your PC. Graphing Calculator is an insanely complete app, thought for helping any need any algebra student may need. It also comes with a smooth interface and a supporting website and email address. Great for Math 51 and Math 60. Gives you different forms of the answer. Very useful. (70) by James Churchill on 17/04/2014 This is a nearly perfect calculator from simple algebra to complex trigonometric/calc. problems, the only downside is that answers with a radical are put into decimal form instead of radical form, (ex. 2r34) other than that this app is nearly perfect! (70) by Devonte Ragsdale on 16/04/2014 I use this app for my stats class and I thrive ! (70) by Dr. Annelida on 16/04/2014 This app is fantastic! My favourite is the graphing function which is really well done
CLASS IX- MATHS SYLLABUS Units: Unit I: Number Systems Real Numbers : (Periods 20) Review of representation of natural numbers, integers, rational numbers on the number line. Representation of terminating/non-terminating recurring decimals, on the number line through successive magnification. Rational numbers as recurring/terminating decimals. Examples of nonrecurring/non terminating decimals such as etc. Existence of non-rational numbers (irrational numbers) such as and their representation on the number line. Explaining that every real number is represented by a unique point on the number line, and conversely, every point on the number line represents a unique real number. Existence of for a given positive real number x (visual proof to be emphasized). Definition of nth root of a real number. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws). Rationalisation (with precise meaning) of real numbers of the type (and their combinations) where xand y are natural numbers and a, b are integers. Unit II: Algebra : Polynomials (Periods 25) Definition of a polynomial in one variable, its coefficients, with examples and counter examples, its terms, zero polynomial. Degree of a polynomial. Constant, linear, quadratic, cubic polynomials; monomials, binomials, trinomials. Factors and multiples. Zeros/roots of a polynomial/equation. State and motivate the Remainder Theorem with examples and analogy to integers. Statement and proof of the Factor Theorem. Factorisation of ax2 + bx + c, a ≠ 0 where a, b, c are real numbers, and of cubic polynomials using the Factor Theorem. Linear Equations in Two Variables (Periods 12) Recall of linear equations in one variable. Introduction to the equation in two variables. Prove that a linear equation in two variables has infinitely many solutions, and justify their being written as ordered pairs of real numbers, plotting them and showing that they seem to lie on a line. Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously. Unit III: Coordinate Geometry : (Periods 9) The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane, graph of linear equations as examples; focus on linear equations of the type ax + by + c = 0 by writing it as y =mx + c and linking with the chapter on linear equations in two variables. Unit IV: Geometry: 1. Introduction to Euclid's Geometry (Periods 6) History – Euclid and geometry in India. Euclid's method of formalizing observed phenomenon into rigorous mathematics with definitions, common/obvious notions, axioms/postulates, and theorems. The five postulates of Euclid. Equivalent versions of the fifth postulate. Showing the relationship between axiom and theorem. 1. Given two distinct points, there exists one and only one line through them. 2. (Prove) Two distinct lines cannot have more than one point in common. 2. Lines and Angles (Periods 10) 1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is180° and the converse. 2. (Prove) If two lines intersect, the vertically opposite angles are equal. 3. (Motivate) Results on corresponding angles, alternate angles, interior angles when a transversal intersects two parallel lines. 4. (Motivate) Lines, which are parallel to a given line, are parallel. 5. (Prove) The sum of the angles of a triangle is 180°. 6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles. 3. Triangles (Periods 20) 1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence). 2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence). 3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence). 4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. 5. (Prove) The angles opposite to equal sides of a triangle are equal. 6. (Motivate) The sides opposite to equal angles of a triangle are equal. 7. (Motivate) Triangle inequalities and relation between 'angle and facing side'; inequalities in a triangle. 4. Quadrilaterals (Periods 10) 1. (Prove) The diagonal divides a parallelogram into two congruent triangles. 2. (Motivate) In a parallelogram opposite sides are equal and conversely. 3. (Motivate) In a parallelogram opposite angles are equal and conversely. 4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. 5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely. 6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and (motivate) its converse. 5. Area (Periods 4) : Review concept of area, recall area of a rectangle. 1. (Prove) Parallelograms on the same base and between the same parallels have the same area. 2. (Motivate) Triangles on the same base and between the same parallels are equal in area and its converse. 6. Circles (Periods 15) Through examples, arrive at definitions of circle related concepts, radius, circumference, diameter,chord, arc, subtended angle. 1. (Prove) Equal chords of a circle subtend equal angles at the centre and (motivate) its converse. 2. (Motivate) The perpendicular from the centre of a circle to a chord bisects the chord and conversely, the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. 3. (Motivate) There is one and only one circle passing through three given non-collinear points. 4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the centre(s) and conversely. 5. (Prove) The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. 6. (Motivate) Angles in the same segment of a circle are equal. 7. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle. 8. (Motivate) The sum of the either pair of the opposite angles of a cyclic quadrilateral is 180° and its converse. 7. Constructions (Periods 10) 1. Construction of bisectors of a line segment and angle, 60°, 90°, 45° angles etc, equilateral triangles. 2. Construction of a triangle given its base, sum/difference of the other two sides and one base angle. 3. Construction of a triangle of given perimeter and base angles. Unit V: Mensuration: 1. Areas (Periods 4) Area of a triangle using Heron's formula (without proof) and its application in finding the area of a quadrilateral. 2. Probability (Periods 12) History, Repeated experiments and observed frequency approach to probability. Focus is on empirical probability. (A large amount of time to be devoted to group and to individual activities to motivate the concept; the experiments to be drawn from real-life situations, and from examples used in the chapter on statistics). Appendix: 1. Proofs in Mathematics What a statement is; when is a statement mathematically valid. Explanation of axiom/ postulate through familiar examples. Difference between axiom, conjecture and theorem. The concept and nature of a 'proof ' (emphasize deductive nature of the proof, the assumptions, the hypothesis, the logical argument) and writing a proof. Illustrate deductive proof with complete arguments using simple results from arithmetic, algebra and geometry (e.g., product of two odd numbers is odd etc.). Particular stress on verification not being proof. Illustrate with a few examples of verifications leading to wrong conclusions – include statements like "every odd number greater than 1 is a prime number". What disproving means, use of counter examples. 2. Introduction to Mathematical Modelling The concept of mathematical modelling, review of work done in earlier classes while looking at situational problems, aims of mathematical modelling, discussing the broad stages of modelling – real-life situations, setting up of hypothesis, determining an appropriate model, solving the mathematical problem equivalent, analyzing the conclusions and their real-life interpretation, validating the model. Examples to be drawn from ratio, proportion, percentages, etc. India's First Online Community Website For CBSE Students. CBSE X, CBSE XII, AIEEE, AIPMT - Papers, Mock Exams, Books, Results and More... IMP NOTE: This website is purely informatory in nature and does not take responsibilty for errors or content posted by community. Disclaimer: cbseportal.com is not at associated with CBSE, For official website of CBSE visit -
Updated and revised edition of the most popular "cheating" guide to statistics! Everything you need to get you through elementary statistics. Expanded t-tests, f-tests and hypothesis testing sections, plus everything from calculating standard deviations to chi-square testsThe book asks why people might believe that numbers 'exist', rather than simply being a concept of our minds? In particular, why should we believe that numbers that consist of the sum of an infinite number of other numbers added together exist? This book presents a convincing argument against the independent 'existence' of such concepts This book comprises many mathematical problems suggested by the author to help the prospective contestants preparing for the Mathematical Olympiad competitions around the world as well as the general audience to learn the concepts and foundations of higher mathematics. These problems are made and tailored in such a way to parallel those used in the past international and national competitions. Game Theory: A Simple Introduction offers an accessible guide to its basic principles and applications. Understand a game matrix, prisoners' dilemma, Nash equilibrium, and the power of asymmetric information. Explore examples looking at free riders, global governance, long-term relationships, competing corporations, advertisers and their customers, along with familiar hawk-dove and chicken game.
Collection Courses The courses section of this website will store curriculum that contains multiple activities or modules, and is meant to span a period of time longer than typically required for a single activity. It may consist of traditional year or semester length courses, online courses that are meant to match a traditional course instruction period, or simply curriculum elements that are lengthier than what one would typically consider a single activity. This course covers the numerical solution of the wave function of the Hydrogen atom, including Schrodinger's equation, Eigenvalue problems, wave equation problems, separation of variables, the radial and angular solution to the wave function, and visualization of the wave function in 3 dimensions. The online course include many runnable models to allow students to explore concepts in the material.
Mathematical Economics is an authoritative collection of the most influential contributions essential to an understanding of this important area of economic science. These seminal papers illustrate the development of the field from its inception in the 19th century up to the present, and exhibit the power of mathematics to lead to new thinking which can illuminate the scientific structures underlying economic arguments. Many of these papers started new fields of economics, influencing deeply the way economists think about their world. They illustrate the extensive range of topics to which mathematics has been applied productively, and show the areas of mathematics which have proved valuable, including functional analysis, linear algebra, algebraic and differential topology, stochastic processes and dynamical systems. They also show the extent to which today's policy analysis rests on yesterday's mathematical economics. Anyone with an interest in economics as a science will find this collection indispensable. The collection is an essential part of any course using mathematical economics.
National 5 Mathematics This Course is valid from August 2013. The National 5 Mathematics Course enables learners to select and apply mathematical techniques in a variety of mathematical and real-life situations. Learners interpret, communicate and manage information in mathematical formUnit assessment overview This document provides general advice on assessment, judging evidence and re-assessment and outlines the approach taken in SQA-produced Unit assessment support packs. Assessors also have freedom and flexibility to produce their own assessments, or use or adapt SQA-produced Unit assessment support packs. In all cases, Unit assessments have to demonstrate competency across all Assessment Standards. Unit Assessment Support As a result of feedback, additional support materials have been published in February 2014 for Outcome 2. A set of questions for the assessment/reassessment of Outcome 2 (reasoning) in National 5 Expressions and Formulae, Relationships and Applications Units has been produced. This is available on the Mathematics Unit assessment support section of the secure site. These have been designed in such a way that the reasoning skills are attached to different operational skills or use a different context or strategy from those used in Unit assessment support package 1. In response to feedback from teachers and lecturers, we made some additions, updates and minor amendments to the Unit assessment support packs for National 4 and National 5 Mathematics in October 2013. These updates are to provide clarification and guidance for teachers/lecturers; particularly across the 'Judging evidence' tables, however the Outcomes and Assessment Standards remain unchanged. The Unit assessment support packs are to assist teachers/lecturers in using their own professional judgement when planning and recording their assessments and while we would encourage use of these latest packs, the previous versions will be accepted for verification purposes. The updated Unit assessment support packs are now available and teachers/lecturers can arrange access to them through their SQA Co-ordinator. This guidance document provides an introduction to new recording tables in the Unit assessment support packs. Conditions of assessment Assessments for Mathematics must be carried out under supervised closed-book conditions. While this is not a change to the conditions of assessment, the UAS packs will be updated to include the term 'closed-book' in order to avoid any uncertainty in the future. Mandatory information Course Specification This explains the overall structure of the Course, including its purpose and aims and information on the skills, knowledge and understanding that will be developed. National Parent Forum Scotland have also produced their Revision in a Nutshell series to help learners to prepare for new National 5 examinations
This book gives a unified approach to the theory concerning a new matrix version of classical harmonic analysis. Most results in the book have their analogues as classical or newer results in harmonic analysis. It can be used as a source for further research in many areas related to infinite matrices. In particular, it could be a perfect starting point... more... With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions... more... This practical and easy-to-understand learning tutorial is one big exciting exercise for students and engineers that are always short on their schedules and want to regain some lost time with the help of Simulink.This book is aimed at students and engineers who need a quick start with Simulink. Though it's not required in order to understand how Simulink... more... Arming readers with both theoretical and practical knowledge, Advanced Linear Algebra for Engineers with MATLAB ® provides real-life problems that readers can use to model and solve engineering and scientific problems in fields ranging from signal processing and communications to electromagnetics and social and health sciences. Facilitating... more... Linear algebra is a fundamental area of mathematics, and is arguably the most powerful mathematical tool ever developed. It is a core topic of study within fields as diverse as: business, economics, engineering, physics, computer science, ecology, sociology, demography and genetics. For an example of linear algebra at work, one needs to look no further book introduces the study of algebra induced by combinatorial objects called directed graphs. These graphs are used as tools in the analysis of graph-theoretic problems and in the characterization and solution of analytic problems. The book presents recent research in operator algebra theory connected with discrete and combinatorial mathematical... more... Self-similar processes are stochastic processes that are invariant in distribution under suitable time scaling, and are a subject intensively studied in the last few decades. This book presents the basic properties of these processes and focuses on the study of their variation using stochastic analysis. While self-similar processes, and especially... more...
Maths Quest General Mathematics Preliminary Course 2nd edition Maths Quest General Mathematics Preliminary Course by Rowland Book Description Maths Quest General Mathematics Preliminary Course Second edition is specifically designed for the General Mathematics Stage 6 Syllabus. This text provides comprehensive coverage of the five areas of study: Financial mathematics, Data analysis, Measurement, Probability and Algebraic modelling. This student textbook offers these new features: * graphics calculator tips throughout the text * a quick and easy way for students to identify formulae that will appear on the HSC examination formula sheet * A CD-ROM that contains the entire student textbook with links to: * interactive technology files; * SkillSHEETS, which assist students to revise and consolidate essential skills and concepts; 2 WorkSHEETS for each chapter, which assist students to further consolidate their understanding Test Yourself multiple-choice questions. The following award winning features continue to be offered in this edition: * full colour with photographs and graphics to support real-life applications * carefully graded exercises with many skill and application problems, including multiple-choice questions * cross-references to relevant worked examples matched to questions throughout the exercises * comprehensive chapter summaries and chapter review exercises with practice examination questions * a glossary of mathematical terms, simply defined * investigations, spreadsheet applications and more. The teacher edition contains everything in the student edition package and more: * answers printed in red next to most questions in each exercise * annotated syllabus information * detailed work programs The teacher edition CD-ROM contains 2 tests per chapter, complete with fully worked solutions, WorkSHEETS and their solutions, and syllabus advice - all in editable World format. Customers who bought this also bought... You might also like... This book helps young learners master fundamental skills necessary for success in school. This title provides more than 90 full-color activities that focus on skills aligned to grade-level curricula. Plus, this book includes access to online timed tests and activities, adding interactive practice of a related skillBooks By Author Rowland Further FP3 is a new title in Oxford A Level Mathematics for Edexcel, a new series that covers the latest curriculum changes and takes a completely fresh look at presenting the challenges of A Level. The author, Mark Rowland, is an experienced teacher who also wrote the other two Further Pure books in this series, FP1 and FP2
I'm a self taught programmer, and although I know many people feel math isn't necessary, I find that in many examples of algorithms I come across talk about (what sounds to be) some pretty complex mathematics. I would love to eventually have a solid understanding of the math that a good, university educated computer scientist should know. I don't really remember any math past algebra 2. With that being where I left off, what should my starting point be? What math topics should I research, and in what order? I'm looking to build a curriculum for myself that will be pretty easy to take on from where I left off and continually learn until I have a similar understanding to that of what a university would provide. Discreet mathematics, linear algebra, set theory, logic (boolean in particular), finite automata and regular languages are the things off the top of my head (and I am as self taught as you are). – Oded♦Mar 7 '12 at 13:16 Hi ThinkingInBits, This question is overly broad and too difficult to answer. There are a number of similar questions however that you can search for and that others have posted as a comment. If you have a more specific question then it will be welcome. – maple_shaft♦Mar 7 '12 at 14:17 I'd suggest looking beyond math itself and into the philosophy of logic, particularly mathematical logic. That will take you further than rote memorization of algorithms and help you find solutions to problems better. – jfrankcarrMar 7 not easy to comprehensively answer your question. It very much depends on what you would like to do with your programming skills. However, as a starting point I would suggest reading the first book of Donald E. Knuth's excellect handbook The Art of Computer Programming. It contains a comprehensive chapter on the mathematical prerequisites for understanding algorithms, and computer science in general. In addition, I feel that a thorough understanding of Boolean algebra and mathematical logic is essential for any programmer.
package for the book, the right to reproduce material for use in his or her own ... answers correctly and a represent the number of application ... DiscoveringAdvancedAlgebra More Practice Your Skills CHAPTER 8 49 The algebra you find in this book won't look quite like the algebra you may have ... DiscoveringAdvanced Algebraalso completes the fully updated ... Selected Answers 796 Glossary 848 Index 860 Photo Credits 872 High School Math Textbook Adoption: Questions & Answers The Bellevue School District has begun the process to adopt a new high school mathematics textbook. ... 1/Geometry/Algebra 2 and Key Curriculum Press DiscoveringAlgebra/Geometry/AdvancedAlgebra. Spring, ... DiscoveringAdvancedAlgebra Textbook 2nd Edition ... graph paper composition book unless instructed otherwise. I expect you to show all your work when appropriate. Homework should be completed and answers checked online prior to class. Late homework is accepted for half credit up I hope that most of the questions in this book deserve an answer. The set of questions is unusually extensive, and is designed to reinforce and deepen students' understanding ... Answer 4.5 In the answers below RA refers to Relational Algebra, TRC refers to Discovering Math Concepts in Business Mathematics Economics and Finance ... • Why was the Treviso Arithmetic an important book? ... understands and applies basic and advanced properties of functions and algebra; understands the general nature and uses of mathematics DiscoveringAdvancedAlgebra Solutions Manual CHAPTER 6 113 ... number of math books sold at the main book-store this week and last week, respectively. The difference between a 11 83 and b ... Answers will vary. LESSON 6.2 EXERCISESbut you may find them at used book stores or web sites, or on eBay. Many of these books are little more than ... ranging from algebra to advanced calculus. ... so students can record their answers as they work. Extensions and extra practices are included at the end of many exercises, ... It is not possible to have a mathematics book entitled . Kidneys. Or worse yet, Life of Fred: ... take a peek at my answers that are given on the page following the questions. ... –10 AdvancedAlgebra. 10–1 Geometry. 11–oon Trigonometry. noon– Calculus This book is the direct result of the author having tutored algebra students and discovering just what they missed ... should find the answers to your algebra difficulties in this book. John McCormick November 5, 2008. ... you should find the answers to your algebra difficulties in this book. DiscoveringAdvancedAlgebra. Key Curriculum Press. Murdock, Jerald; Kamischke, ... ... our answers to the same problem to the same number of decimal places, ... Referring back to the book, you would want to identify three: (1) the Industrial Revolution; ... ANSWERS TO MATCH THESE SOCIAL SCIENTISTS WITH THEIR CONTRIBUTIONS 1. d Gerhard Lenski: status inconsistency 2. i C. Wright Mills: power elite For each book on the list, note characters, plot, ... Left to Tell: Discovering God Amidst the Rwandan Holocaust Immaculee Ilibagiza ... All students entering Advanced Pre Algebra are to purchase the workbook: SPECTRUM MATH published by After discovering a shared love of art, two sixth grade students are ... Advanced Preparation by Teacher: 1. ... Using their notes and book, Algebra 2 students will graph a parabola (type of conic section) with an ALGEBRA Y TRIGONOMETRIA ... Asking "What makes the math sciences so central?" he answers by quoting Galileo: "The great book of nature can be read only by those ... interaction of the ocean, wind, and boat. They then applied advanced numerical algorithms to solve these equations on high ... courses like college algebra, discrete mathematics, probability and statistics, ... majority of my time discovering what other researchers have determined on the topic of using ... book. These answers were so obscure that it was not even likely they were from more advanced Discovering wavelets / Edward Aboufadel and Steven Schlicker. p. cm. ... The major benefit of this book is that it presents basic and advanced con- ... Answers to selected problems arc provided in an appendix. Pixel Images. and. Advanced Extension Award (AEA) in Mathematics or the Sixth Term Examination Paper (STEP) ... which mathematics can provide clear cut answers to nature, the universe and life. ... Discovering Mathematics: The Art of
Online Distance Learning Practical math applications are featured in this course. You will develop skill in whole numbers, fractions, decimals, percents, ratio and proportion, linear measurement, basic geometry and volume. Both the imperial and metric systems are utilized. This course will help students assess computer skills and prepare for online learning. Topics include basic computer knowledge, e-mail and the internet. You need to be able to use a mouse and load a CD Rom to begin this course
Summary The Eighth Edition of this highly dependable book retains its best features-accuracy, precision, depth, and abundant exercise sets-while substantially updating its content and pedagogy. Striving to teach mathematics as a way of life, Sullivan provides understandable, realistic applications that are consistent with the abilities of most readers. Chapter topics include Graphs; Polynomial and Rational Functions; Trigonometric Functions; Analytic Trigonometry; Analytic Geometry; Counting and Probability; A Preview of Calculus; and more. For individuals with an interest in learning Precalculus as it applies to their everyday lives. Table of Contents (All chapters end with a Chapter Review and a Project at Motorola). Preface. List of Applications. 1. Graphs. The Viewing Rectangle. Using a Graphing Utility to Graph Equations. Using a Graphing Utility to Locate Intercepts and Check for Symmetry. Using a Graphing Utility to Solve Equations. Square Screens. Using a Graphing Utility to Graph Inequalities. Using a Graphing Utility to Solve Systems of Linear Equations. Using a Graphing Utility to Graph a Polar Equation. Using a Graphing Utility to Graph Parametric Equations.
About: Quadratic Equations: Proficiency Exam Metadata Name: Quadratic Equations: Proficiency Exam ID: m21920 Language: English (en) Summary: This module is from Elementary Algebra</link> by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method. This module contains the proficiency exam for the chapter "Quadratic Equations".
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). tha
Shop by Price Get Newsletter Algebra The AskDrCallahan Algebra DVD course is perfect for homeschool or other self driven students. The course provides a solution for parents and students looking for a solution to learning algebra and preparing them for geometry, the ACT, SAT, and future math success. The student is guided through the Elementary Algebra textbook by Harold Jacobs with video lessons covering each topic and working examples as needed. Word problems are walked into using real world examples and simple logic. Course includes everything needed for mom and student alike. The course is taught in a light hearted style, focused on understanding the concepts of algebra and not a ton of math-speak. Printer Teachers Guide (SKU 1211): Printed copy of the AskDrCallahan Algebra Teachers Guide which includes the tests, the test grading guide, and the syllabus. The guide is contained on the CDs with SKU 1210 (DVDs) but this offers a printed copy of the same. You may also download the Teacher's Guide. Other Common Questions How do I konw if my student is ready for algebra? We get this all the time. Many of you are wanting to start in a pre-algebra to get your student "up to speed". We look at this is bit differently. If your student can understand the following, they are ready for algebra X + 5 = 10 and figure out what X is. Plot a point on a graph. Understands basic addition, multiplication, subtration, and division. (No, they do not have to know "math facts" such as what 6 times 8 equals without thinking about it.) Basic fractions. Do they know what 1/4 of a cup is and how to find it? What about 2/4ths of a cup? Really, we are not kidding. Most prealgebra is a repeat of what they have done before. So jump into algebra and take it slow. It has all the solutions. Sometimes they are in detail, and other times it is just the answer. Here is a sample of the first 2 chapters of the Jacobs Solution Manual. We know that some homeschool material works out everything in detail, and this does NOT do that. However, the solutions manual offers all you will need. When detailed answers are not given, it is because they are pretty obvious. On the rare occassion that your student (or you) does not understand how the solution was found, we will help you via our support. Over the years, we have had very few problems from parents or students related to this issue. Can my son/daugher do this course on their own? Yes. We know, we homeschool too. There is parent overload. So we have designed these courses so that you can either do it with them or have them do it on their own. For our children, they do everything on their own until they have a question. Furthermore, I would suggest that you DO have them work it on their own. Again, you are preparing for college and life, and you want to teach them to learn on their own as well as to have personal responsibility.
Jump to Overview Presents a step-by-step approach to do all kinds of business-math problems, including basic arithmetic, algebra, percentages, and conversions; specific, real-life applications such as figuring discounts, calculating simple and compound interest, reading balance sheets and income statements, and preparing graphs and charts; and, more.
... Show More failure rates. A large number of real-world examples and problems enable students to create mathematical models that will help them understand the world in which they live. The focus is on those topics that are essential to the study of calculus and these topics are treated in depth. Linear, exponential, power, and periodic functions are introduced before polynomial and rational functions to take advantage of their use to model physical phenomena. Building on the Consortium's Rule of Four: Each function is represented symbolically, numerically, graphically, and verbally where appropriate
Developed by Dr. Abdelkader Dendane, a lecturer at the United Arab Emirates University, the Analyze Math website is designed to provide curious minds with access to in-depth tutorials on such topics as logarithmic... A course designed to familiarize high school and beginning college mathematics teachers with advanced algebra content and teaching strategies. The approach stresses modeling and solving real world problems and develops... With a development team that includes several well-regarded mathematicians and other such folk, the S.O.S. Mathematics website is a high-quality resource for persons who might find themselves in need of a bit of... From abelian group to zero divisor (definitions), and from Artin-Wedderburn theorem to Wedderburn's theorem, this site provides concise explanations of complex concepts in abstract algebra. Provided by Professor John A.... "In the Classroom" highlights how some schools and organizations use Mathematica extensively in their curricula. The section on "Collaborative Initiatives" illustrates how businesses have teamed up with Wolfram Research...
Summary: This text is for a one-term course in intermediate algebra, for students who have had a previous elementary algebra course. A five- step problem-solving process is introduced, and interesting applications are used to motivate students. Coverage progresses from graphs, functions, and linear equations to sequences, series, and the binomial theorem. New to this edition are sections on connecting concepts, study tips, and exercises designed to foster intuitive problem so...show morelving. Bittinger teaches at Indiana University; Ellenbogen at Community College of Vermont. ...show less 2 CD'S INCLUDED GOOD CONDITION ...show less $4.82 +$3.99 s/h Good Daisybooks Layton, UT Hardcover Good 0321557182 includes unopened disc, pages are clean and free of markings, light wear to corners and edges, ships same day or next. $4.898.25 +$3.99 s/h Good Bookmans AZ Tucson, AZ 2009 Hardcover Good Satisfaction 100% guaranteed. $19.12 +$3.99 s/h Good citywbooks Salt Lake City, UT Clean pages with general wear from reading and storage. If the dust jacket is present it may have curling, wear, or small tears on the edges. Some books have a bookstore stamp inside cover. Quick resp...show moreonse! ...show less $19.69 +$3.99 s/h Good invisibledog Salt Lake City, UT 0321557182 Unmarked text. CD-ROM still sealed. $20.95 +$3.99 s/h Good LotsofBooks Nashville, TN No comments from the seller $31.48
0130107050 9780130107053 A Friendly Introduction to Mathematical Logic:For a one-quarter/one-semester, junior/senior-level course in Mathematical Logic.With the idea that mathematical logic is absolutely central to mathematics, this tightly focused, elementary text discusses concepts that are used by mathematicians in every branch of the subject a subject with increasing applications and intrinsic interest. It features an inviting writing style and a mathematical approach with precise statements of theorems and correct proofs. Students are introduced to the main results of mathematical logic results that are central to the understanding of mathematics as a whole. Back to top Rent A Friendly Introduction to Mathematical Logic 1st edition today, or search our site for Christopher C. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Prentice Hall.
Introduction to Maple 16: See What's New Can't see the video? Update to the latest Adobe Flash Player, or use an HTML5 compatible browser. Description Maple 16 is a major new release of Maple, the technical computing software for mathematicians, engineers, and scientists. With Maple 16, Maplesoft introduces new tools and techniques in its Clickable Math collection, setting new standards for ease of use in mathematical software and providing new, innovative ways to explore mathematics. Of course, the interface is only one of the many areas of Maple that was enhanced in Maple 16. Substantial improvements were made across the entire product, in mathematical algorithms, visualization, programming language, engineering tools, documentation, education features, authoring tools, and more. In this webinar, we'll explore many of the new features of Maple 16.
Elementary and Intermediate Algebra - Worksheets - 5th edition Summary: This workbook provides one worksheet for each section of the text, organized by section objective. Each worksheet lists the associated objectives from the text, provides fill-in-the-blank vocabulary practice, and exercises for each objective. 0321599330 Instructors edition! Item has some cover wear but otherwise in good condition!!Used texts may not include supplemental matieral.95 +$3.99 s/h VeryGood RF Ventures Cedar Hill, TN "Book is in very good condition. Some answers written in pencil first 11 pages, No writing the rest of book."
Bharath Sriraman Dept. of Mathematical Sciences The University of Montana Abstract The interaction of the history of mathematics and mathematics education has long been construed as an esoteric area of inquiry. Much of the research done in this realm has been under the auspices of the history and pedagogy of mathematics group. However there is little systematization or consolidation of the existing literature aimed at undergraduate mathematics education, particularly in the teaching and learning of the history of mathematics and other undergraduate topics. In this monograph, the chapters cover topics such as the development of Calculus through the actuarial sciences and map making, logarithms, the people and practices behind real world mathematics, and fruitful ways in which the history of mathematics informs mathematics education. The book is meant to serve as a source of enrichment for undergraduate mathematics majors and for mathematics education courses aimed at teachers. Keywords: History of Mathematics; Mathematics Education; Undergraduate Mathematics Education; History of Analysis; History of Geometry
Calculus I (Video) by integralCALC 302 Videos Overview This course helps you build a strong foundation for calculus by reviewing vital calculus concepts. Taught by Krista King, a calculus tutor with 10 years experience, this course makes calculus accessible to everyone with her simple, step-by-step tutorials. Topics include Pre-Calculus, Limits & Continuity, Derivatives, Applications of Derivatives and Integrals. Calculus I course serves dual purpose of being a great refresher course for college students and a perfect gateway course for high school students. Take Advantage of BenchPrep's Features Now Related Courses From Sequences, to Gamma and Beta Functions, Calculus by Schaum's Outlines offers a broad overview of Calculus, designed to better acquaint the student with the subject using concise, easy to understand explanations.… Learn more With 217 practice questions, 2 mini-tests, and 1,732 flashcards, Schaum's Calculus Practice Questions is a vital tool in helping students better their Calculus skills. All the material is paired with concise,… Learn more Patrick JMT has been teaching mathematics at the college level for 8 years and has taught at Vanderbuilt University and at the University of Louisville. This course contains 251 video lessons in which Patrick guides you… Learn more Patrick JMT has been teaching mathematics at the college level for 8 years and has taught at Vanderbuilt University and at the University of Louisville. This course is the next step once students have mastered the… Learn more Covering everything calculus from Limits and Continuity to Parametric Equations and Polar Coordinates, Calculus I by NJIT delivers you the essential understanding you need to strengthen your mathematical skill. In this… Learn more
Algebra I am getting ready to start an Algebra class for the first time and I was wondering what challenges people have with learning and using algebra concepts. Also what are the best ways to over come math anxiety? The best way? Learn the language of algebra. Tuesday, March 20, 2007 at 9:02am by jeff SHAY THE MATH QUESTION IS IT ALGEBRA , PRE ALGEBRA' OR GEOMETRY? Quadriatic functions. so algebra i think. Wednesday, January 17, 2007 at 7:09pm by ROSA what does pre-algebra mean?? Pre-Algebra is basically preparing you for Algebra. Pre-Algebra teaches you Order of Operations, Properties of Numbers, Rational and Irrational Numbers, Exponents, PEMDAS, ect. Monday, August 25, 2008 at 9:11pm by Delilah College Algebra is this really algebra, im doing this now in freshman year algebra 2 Monday, February 21, 2011 at 1:06am by BOSS Algebra 1A How is algebra a useful tool? what concepts investigated in algebra can be apply to personal and professional life? I need help answering this question. Please help? Sunday, November 30, 2008 at 7:34pm by Julissa algebra 2 thats part of algebra 2 thats easy were doing that now in pre algebra Wednesday, April 29, 2009 at 1:14pm by Tanisha Algebra (Intermediate) Tutors can better help you when they know they're working with the same student. You might try something like this: Algebra (1), Algebra (2), and so on. Also -- you are more likely to get assistance if you tell us what you know and what you don't understand about your problem. Monday, October 11, 2010 at 12:26pm by Ms. Suewhat does pre-algebra mean?? pre algebra is like a bunch of math that comes before algebra in middle school. Monday, August 25, 2008 at 9:11pm by Grace Algebra 2 In Kentucky, where i am from, we do algebra and algebra two before pre-cal and calculus. I didnt realize that the problem could be solved in multiple ways and i apologise. But the solution should be done without calculus. Thursday, February 24, 2011 at 10:33pm by Anon algebra Once again did you click on the "answer" part of examples #6 and #7 ? Tuesday, December 1, 2009 at 10:30pm by Reiny Algebra-still need some help Homework Help Forum: Algebra Posted by Jena on Thursday, February 3, 2011 at 7:32pm. Find the domain of the function. f(x)=(sqrt x+6)/(-2x-5) Write your answer as an interval or union of intervals. Algebra - David, Thursday, February 3, 2011 at 7:42pm (-6,-5/2)and(-5/2,... Thursday, February 3, 2011 at 8:43pm by Jena algebra 1 at a certain high school,350 students are taking algebra. the ratio of boys to girls taking algebra is 33:37. How many more girls are taking algebra than boys? - How can you write a system of equations to model the situation? - Which equation will you solve for a variable in ... Sunday, January 27, 2013 at 3:24pm by lucy 7th grade There are at least 5 more than twice as many students taking algebra 1 than taking algebra 2. If there are 44 students taking algebra 2, what is the least number of students who could be taking algebra 1. Show all work Thursday, November 20, 2008 at 8:37pm by lee Algebra Nor was I in my algebra class of 1943. This COULD be a case of changing the rules (as has been done with all the SI units). A micron isn't a micron anymore (:(]. In fact my algebra teachers said, "DON'T forget there is a negative root of the square root of 4." Sunday, January 4, 2009 at 5:46pm by DrBob222 College Algebra Thank you very much. I'm not doing very good understanding this algebra right now so I will definately have other questions tonight. I'm working on getting a tutor here in my town because right after this class is over Sunday, I go into Algebra two. Wednesday, November 18, 2009 at 7:41pm by LeAnn/Please help me Algebra The average mark on a test in an algebra class is 80. If the two lowest scores of 34 and 48 are not counted, the remaining scores would average 83. How many students are in the algebra class? Thursday, October 28, 2010 at 7:52pm by Rocky algebra x = 4 is the answer, if you take the positive square roots. There may be other answers if you take the negative square root on one or both sides. I got that by trial and error, not by using algebra. The algebra got too messy. Tuesday, July 22, 2008 at 11:00pm by drwls pre-algebra kk my teach told me to describe and draw things that are about pre-algebra and she told me to write what pre-algebra is... so can anyone help me?? Monday, August 25, 2008 at 9:03pm by damainmind Math Check this site. If it doesn't help you, please post one or two of your problems. We'll be happy to show you how to find the solutions. Tuesday, July 29, 2008 at 9:27pm by Ms. Sue Algebra I am having a problem with solving a composite function in Algebra 1 How do I solve f(x) = -2x + 1 and g(x) = 4x? Don't I need more info? No, that is all you need. I it unclear what you want to do here. see Wednesday, July 18, 2007 at 9:45pm by Alec collage algebra 1 no need to throw a hissy-fit. I also wondered why somebody claims to have a "college Algebra" question and can't spell 'college' Besides, this is at most a grade 9 type algebra question. Anyway, why don't you substitute the value of x given in the second equation into the ... Monday, January 5, 2009 at 10:55pm by Reiny algebra An algebra book weighs 6 oz less than twice as much as a grammar book. If 5 algebra books weigh the same as 8 grammar books, how much does an algebra book weigh? Wednesday, October 19, 2011 at 12:00am by AnonymousAlgebra Study this site. If you have a specific question, please post it, and we'll try to help you. Saturday, January 11, 2014 at 7:31pm by Ms. Sue Algebra 1 Just a note: this problem is normally well above the ability of algebra I students. Thursday, June 26, 2008 at 8:28pm by bobpursley pre-algebra(algebra) I might need more help i am still working on my homework Wednesday, October 3, 2007 at 9:27pm by Stacy Algebra 2(check pt 4) I am so stuck on these same exact problems on my Algebra 2 test. Saturday, July 28, 2007 at 3:55pm by Chels
Full course description This course is a review of Basic Arithmetic skills that serve as a prerequisite for placement into and success in pre-college and college-level algebra courses. In this course, primary emphasis will be placed on fundamental operations with whole numbers, fractions, decimals, and integers. Other topics covered include proportions, percentages, representations of data, geometric figures, and measurement. Students who should take this course include: those that have an interest in brushing up on arithmetic skills prior to taking an upcoming placement test or those that have not had math in many years and want to review foundational skills and concepts. This course provides free digital access to all required materials including a student workbook, lesson videos, and online homework practice and assessment. The course instructor recommends purchase of a textbook or other course materials. Please see the details below. Donna Gaudet, Ed.D. Professor of Mathematics Donna Gaudet is an instructor of mathematics at Scottsdale Community College in Scottsdale, Arizona. She received her Ed.D in Instructional Technology and Distance Education from Nova Southeastern University in 2006. She also earned an M.S. in Applied Mathematics from the University of Colorado at Colorado Springs (1991) and a B.A. in Mathematics from the same school (1989). Donna has used technology actively in her courses since the mid-1990s and her current interest is in the design and delivery of hybrid and online learning experiences and courses.
Boost your students understanding of Saxon Math with DIVE's easy-to-understand lectures! Each of the 120 lesson concepts and 12 investigations in Saxon Math's Geometry textbook is taught step-by-step on a digital whiteboard, averaging about 10-20 minutes in length; and because each lesson is stored separately, you can easily move about from lesson-to-lesson as well as maneuver within the lesson you're watching. After the lesson, students complete the 30-question Problem Set in the Saxon text; a few problems in the set come from the new concept, however most review previous lessons. Each problem is cross-referenced back to the lesson where it was taught, allowing students to quickly review any concept This course covers all topics in a high school geometry course, including perspective, space, and dimension associated with practical and axiomatic geometry. Students learn how to apply and calculate measurements of lengths, heights, circumference, areas, and volumes, and will be introduced trigonometry and transformations. Students will use logic to create proofs and constructions, work with key geometry theorems and proofs, and use technology such as spreadsheets, graphing calculators, and geometry software.
The predictors and consequences of eighth grade algebra success ProQuest Dissertations and Theses, 2011 Dissertation Author: Frank Nicholas Stephany Abstract: The nature of Algebra, and the essential fundamentals that children master through studying Algebra serve as an essential building block of further mathematical studies. This study underscores the importance of success in Algebra. TABLE OF CONTENTS Dedication………………………………………………………………………… …ii Acknowledgments……………………………………………………………….. …iii List of Tables……………………………………………………………………... …v List of Figures…………………………………………………………………… …vii Abstract………………………………………………………………………… …viii Chapter One: Introduction…………………………………………………… …1 Chapter Two: Review of the Literature……………………………………... …17 Chapter Three: Methodology…………………………………………………. …55 Chapte r Four: Results……………………………………………………….. …68 Chapter Five: Conclusions………………………………………………… …133 References……………………………………………………………………… …149 Appendices Appendix A: Interview Letter of Introduction/Explanation………... …154 Appendix B: Counselor Interview Protoco l…………….......……….…156 v LIST OF TABLES Table 1. Cohort Selection Criteria……………………………………………. …60 Table 2. Percentage of Students in Eighth Grade Algebra…………………… …81 Table 3. Total Students by Parent Education Level………………………….. …83 Table 4. Participation in F ree and Reduced Lunch Program…………………. …85 Table 5. Participation in GATE………………………………………………. …86 Table 6. Math Course Selection for GATE Students in Eighth Grade……….. …86 Table 7. Special Education Student Participation…………………………….. …87 Table 8. 504 Accommo dation Plan Student Participation……………………. …87 Table 9. Average Math CST Scaled Scores by Elementary School………….. …90 Table 10. Average CST Scores by Intermediate School………………………. …91 Table 11. Average 5th Grade Math and Algebra I CST Scales Scores………... … 93 Table 12. Average 7th Grade Math and Algebra I CST Scaled Scores……….. …94 Table 13. Population of LAHS Graduating Classes 2008 - 2010………………. …97 Table 14. Total Students by Parent Education Level………………………… …100 Table 15. Weighted, Cumulative GPA (2008 - 201 0 LAHS graduates)………. …102 Table 16. Participation in the Free and Reduced Lunch Program……………. …103 Table 17. Participation in GATE……………………………………………... …103 Table 18. Math Course Selection for GATE Students in the Eighth Grade….. …104 Table 19. Special Edu cation and 504 Accommodation Plan Student………… …104 Participation Table 20. Cohort Selection Criteria, Algebra I Variables……………………. …107 vi Table 21. Math SAT Averages by Cohort……………………………………. …108 Table 22. Correlation Coefficients between Math SAT and Algeb ra I CST…. …108 Scaled Scores Table 23. Algebra II CST Scores and Averages by Cohort………………….. …110 Table 24. Correlation Coefficients between Math SAT and Algebra I CST… …110 Scaled Scores Table 25. Grade Level of Last Math Course (shown in percentages)… ……... …114 Table 26. Average Cohort's Final Mathematics Course and Grade Level…… …115 Table 27. Accelerated Cohort's Final Mathematics Course and Grade Level …117 Table 28. High Achiever Cohort's Final Mathematics Course and Grade…… …118 Level Table 29. G rade Level of Final Science Course (shown in percentages)…….. …120 Table 30. Average Cohort's Final Science Course and Grade Level………… …121 Table 31. Accelerated Cohort's Final Science Course and Grade Level……... …123 Table 32. High Achiever Cohort's Final Science Course and Grade Level….. …124 vii LIST OF FIGURES Figure 1. LAUSD Demographics…………………………………………….. …57 Figure 2. Excerpt from the 6th Grade Math Recommendation Form………... …72 Figure 3. Multiple Measures for Pre - Algebra Identification: Grade 6………. …74 Figure 4. Math Course Sequence, 6th to 7th Grade………………………….. …76 Figure 5. Math Course Sequence, 7th to 8th Grade………………………….. …77 Figure 6. Math Course Sequence, 8th Grade to 12th Grade…………………. …78 Figure 7. Percentage of Students in Eighth Grade Algebra… ……………….. …82 Figure 8. Ethnicities of the Overall Eighth Grade Student Population………. …82 Figure 9. Parent Education Level of Students (shown in percentages)……… …84 Figure 10. Ethnicity of Students that took Eighth Grade Algebra I…………… …98 (2004 - 2006) Fig ure 11. Ethnicity of the Overall Student Population that Graduated………. …99 High School (2008 - 2010) Figure 12. Parent Education Level of Students (shown in percentages)……... …101 viii ABSTRACT This study examined predictors of success in eighth grade Algebra a nd the consequences of that success in high school. There has been a nationwide push to enroll more eighth grade students in Algebra in order to allow access to advanced mathematics in high school. The increase of students enrolling in Algebra has not on ly Using the Los Amigos Unified School District as a model for this case study, it was found that fifth and seventh grade CST sco res This study lends supportive evidence to the belief that success in Algebra I is of more importance than the g rade level at which Algebra I is taken. The nature of Algebra, and the essential fundamentals that children master through studying Algebra serve as an essential building block of further mathematical studies. This study underscores the importance of suc cess in Algebra. 1 CHAPTER ONE INTRODUCTION The ability of the United States to adequately educate and prepare its students in the use of quantitative ideas is critical to its standing as a world leader in economics, technology and scientific innovation. Hi story is full of examples of civilizations that have thrived and flourished as a direct result of their ability to understand and apply mathematical concepts. "Leading societies have commanded mathematical skills that have brought them advantages in medic ine and health, in technology and commerce, in navigation and exploration, in defense and finance and in the ability to understand past failures and to forecast future developments" (NMAP, 2008, p. 1). An understanding of mathematical principles is necess ary in order to preserve the quality of life to which the citizens of the United States have become accustomed. "Although the need for mathematics within science and technology fields is significant, its role goes beyond career preparation; mathematics re asoning is an indispensable tool for informed participation in a democracy" (Gine & Kruse, 2007, p. 1). Mathematical preparation is also a vital component of the United States' ability to adequately prepare a workforce that is able to keep pace with and a dapt to the challenges of the 21 st century global economy. Job growth in the mathematics, science and engineering workforce are outpacing overall job growth and that trend is sure to continue. Mathematic proficiency is an essential element if the United States is to maintain its position as a world leader. 2 Since World War II, the government and educators of the United States have recognized the need for its citizens to obtain mathematic proficiency and have made it the focus of a number of reports, reform s and initiatives designed to raise mathematical achievement in the nation's schools. Herrara & Owens (2001) suggest that the 1957 launch of the Soviet satellite Sputnik helped to create the perception that the United States trailed the Soviet Union in sc ience and technology. Sputnik helped to refocus the nation's attention on math and science education as an integral component to success and security. The result was an educational reform known as the "New Math Movement." This movement of the 1960s focu sed predominately on preparing students for higher level courses such as calculus and analytical geometry mathematics at the university level. The discovery and problem solving techniques of the New Math Movement eventually gave way to the "back to basics " era of the 1970s in which educators employed procedures and principles backed by behavioral psychology (Herrara & Owens, 2001). Students were drilled in the basics and learned the procedures for solving complex mathematical problems. During both decade s, neither reform resulted in substantial gains in student mathematics achievement. In 1983, the National Commission for Excellence in Education brought renewed attention to the need for adequate mathematics preparation in its Nation at Risk report. The d evelopment of standards by the National Council of Teachers of Mathematics (NCTM) was one of the outcomes of this report. "Mathematical curriculum standards refer to the set of learning goals articulated across grades that 3 outline the intended mathematics content and process goals at particular points in time throughout the K – 12 mathematics program" (Reys, Reys, Lapan, Holliday & Wasman, 2003, p. 75). The use of standards - based curricula has now become the accepted standard for mathematics instruction a cross the nation. Many studies have shown that student achievement in mathematics has significantly increased with the use of a standards - based curriculum, including students with varying ability levels and backgrounds (Briars, 2001; Griffin, Evans, Timms & Trowell, 2000). Currently state and federal laws require that all public schools teach to approved standards and that students' proficiency of the standards is assessed. Despite advances made in content and pedagogy, student achievement in mathematics remains dismal. According to the 2007 National Assessment of Educational Progress (NAEP) scores, only 39% of students in grade 8 and only 23% of students in grade 12 are at or above the proficient level (Lee, Grigg & Dion, 2007). Several measures of stud ents' mathematical understanding show that American students do not fare well compared to the youth of other industrialized nations. In the Trends in Mathematics and Science Study (TIMSS), 38% of the fourth graders in Singapore scored at an advanced level while only 7% of the fourth graders in the United States did so (NMAP, 2008). Another study by Baldi, et al. (2007) showed that American 15 year old students' understanding of math literacy and problem solving ranked 25 out of 30 among developed nations. Even more alarming are 2003 NAEP scores that showed many eighth grade students lack the fundamental skills required for algebra including familiarity with fractions and word 4 problems (Braswell, et al., 2005). These examples illustrate a much larger prob lem: students in American public schools are not acquiring the mathematical knowledge necessary to maintain America's position of leadership in the global community. Algebra has become a major focus for increasing students' mathematical achievement. The N CTM (2008) describes algebra as "… a way of thinking and a set of concepts and skills that enable students to generalize, model and analyze mathematical situations. Algebra provides a systematic way to investigate relationships, helping to describe, organ ize and understand the world" (p. 1). Algebra is seen as a gatekeeper course that teaches students skills that will enable them to be successful in higher level mathematics courses. Students who complete high advanced math courses such as Algebra II have more success in high school and college mathematics courses and eventually have a higher earnings potential (NMAP, 2008, p. xii). "Algebra offers a way to generalize mathematical ideas and relationships, which can then be applied in a wide variety of mat hematical and nonmathematical settings" (NCTM, 2008, p. 1). Thus, algebra is seen as an integral component to raising students' math achievement. It was observed on both the 2003 and the 2008 California Standards Test (CST) in mathematics that students d emonstrated a marked decline in mathematics achievement during their intermediate school years; close to the same time that they enter or are preparing to enter their first formal algebra course (EdSource, 2009). Many educators are concerned that poor mat h achievement during students' middle school years will discourage students in their first algebra course contributing to less future success in 5 and a general dislike for mathematics (NMAP, 2008). As one of the primary and most important mathematical skil l sets, algebra equips students to pursue higher mathematics contributing to greater mathematical achievement. There has been a significant ground swell of support to teach algebra in the eighth grade. The emphasis on algebra in the eighth grade was broug ht to national attention when President Clinton declared it a national goal in Goals 2000. Many states such as California now insist that students take Algebra in the eighth grade by employing assessment and accountability measures. The body of literatur e suggests that earlier algebra completion will result in greater chances of taking advanced mathematics courses, higher mathematical achievement and greater likelihood of college admission and graduation (Loveless, 2008). Along these lines, educators and policy makers have pushed for students to take algebra before entering high school. Middle school mathematics course selection has real academic consequences: often placing students on a track that predicts their course placement at the high school level (Oakes, Gamoran & Page, 1992). "Eighth grade mathematics generally is considered to be the gatekeeper to advanced mathematics learning, entry into physical science courses, and post secondary opportunities" (Wang & Goldschmidt, 2003, p. 14). Current res earch suggests that the earlier algebra has been taught, the greater the students' mathematical achievement and policy makers have decided that algebra should be taught in the eighth grade. Student access to and success in algebra has been shown to differ across ethnic and gender lines. The number of mathematics courses taken differs according 6 to ethnicity and gender. "Girls and underrepresented minorities, especially African Americans and Hispanics, tended to take few mathematics courses" (Wang & Goldsch midt, 2003, p. 3). In 2003, only 23% of black students and 26% of Latino students took Algebra in the eighth grade limiting most of these students' chances of completing Algebra II by the end of high school (EdSource, 2009; Tietelbaum, 2003). Students ta king fewer mathematics courses are often at a disadvantage when applying for college admission. The NMAP Final Report (2008) demonstrated that mathematics success is related to college success which in turn is related to higher earnings and a higher quali ty of life. Thus, access to algebra courses for all students in eighth grade is an equity issue. "The push for universal eighth - grade algebra is based on an argument for equity … democratizing eighth - grade algebra promotes social justice" (Loveless, 2008 , p. 3). While the immediate objective of increasing access to algebra for eighth grade students has succeeded, the long term goals of increasing student math achievement and access to higher level mathematics have yet to be realized. The math achievement of California students, as measured by the CST in 2003 and 2008, has shown that even though algebra enrollment in the eighth grade increased by 43% over five years, nearly the same percentage of students is scoring at or above proficient level (EdSource, 2009). Thus, while student participation in algebra at the eighth grade level has increased, student success has remained fairly stagnant. The 2007 NAEP scores showed that from 2000 to 2007, the achievement of eighth grade students in advanced classes su ch as Geometry and Algebra II has decreased by four 7 points (Lee, Grigg & Dion, 2007). Across gender, ethnicity, linguistic background and learning ability, the increase in the number of students taking Algebra in the eighth grade does not yield greater st udent success in math. Data also show that many eighth grade algebra students repeat algebra in their first year of high school. As EdSource claims more than one - third of students taking Algebra in their ninth grade year are repeating the same course tha t they took in eighth grade (EdSource, 2009). Loveless (2008) attributes these concerning trends to students that are inappropriately accelerated and subsequently misplaced in Algebra courses in the eighth grade. Eighth grade students that are misplaced in advanced algebra courses were shown to score significantly lower than their peers on the 2007 NAEP (Loveless, 2008). While the increasing number of eighth grade students taking algebra courses is encouraging, their lack of success is cause for concern. Schools must weigh the pressure to accelerate all students to take algebra in the eighth grade, keeping in mind the mathematic preparation and scholastic well being of the child. Some scholars worry that a continuous lack of success in math can have the effect of convincing students that they are unable to learn the material and will create general dislike of mathematics (EdSource, 2009). Loveless (2008) suggests that NAEP scores show that eighth grade students misplaced in advanced algebra courses are a ctually seven grade levels of mathematical achievement below their appropriately placed peers in the same mathematics courses. Additionally, many high schools will not recognize Algebra courses taken at the middle school level, questioning its rigor and f orcing students to repeat the course in the ninth grade 8 year (EdSource, 2009). The push to accelerate students to take Algebra in eighth grade may result in unintended consequences for student achievement while not progressing towards the primary goal of putting students on track to take higher level mathematics courses in high school. The opportunity for an eighth grade student to take algebra means nothing if that student has not been adequately prepared in the prerequisite skills necessary to be success ful in that course. The increase of students taking algebra courses has resulted in a situation in which students are taking courses for which they are not prepared. The NCTM (2008) remarks: Only when students exhibit demonstrable success with prerequisi te skills – not at a prescribed grade level – should they (students) focus explicitly and extensively on algebra, whether in a course titled Algebra I or within an integrated mathematics curriculum. Exposing students to such coursework before they are rea dy often leads to frustration, failure and negative attitudes towards mathematics and learning (p. 1) The California Mathematics Framework explicitly states that students should be prepared to study algebra by the eighth grade. However, student learning must be the primary goal, not course completion. The practice of placing all eight grade students in algebra may not be the best way to increase student math achievement. Given that preparation is crucial to success in algebra courses, schools and distric ts are looking to find effective means of assessing student algebra preparedness in order to place students in the most appropriate math course in the eighth and ninth grades. Students placed in courses that are appropriate for their skill level will bene fit more than students who are accelerated and misplaced (Loveless, 9 2008). Therefore, a school's ability to adequately assess students' algebra preparedness and to place students in challenging yet appropriate math courses may ultimately determine that sc hool's math achievement. The numerous, current predictors for a students' success in algebra are as follows: standardized test scores, academic marks, placement test results and teacher recommendations are commonly used to place students in the appropriat e math class. Not only must schools and districts be able to rely on accurate and consistent predictors of student success in algebra, they should understand the effect that taking algebra in the eighth grade can have on students' college readiness. State ment of the Problem Achievement of the primary goal of increasing student math achievement and a greater aptitude for problem solving and technical skills is critical to the United States' ability to meet the upcoming challenges of the 21 st century. The p ush for algebra in the eighth grade has reached a sort of academic crossroads, carefully attempting to weigh the increased likelihood of future success by taking advanced math courses with the preparation necessary to enroll in such a course. Teachers and administrators in public schools are under pressure to enroll eighth grade students in algebra courses from state and federal governments; however, increased enrollment in eighth grade algebra has contributed to lower student math achievement and more stu dents repeating the course. Many observers feel that these trends are the consequences of eighth grade students misplaced in advanced algebra courses without adequate preparation. In order to realize the greater goals of accelerating 10 students to take Alg ebra in eighth grade, schools must ensure that students are adequately prepared for Algebra prior to placing students in courses that they are destined to fail and repeat. The teachers and administrators of individual schools must make math course placemen t decisions that affect their students' future, their school's achievement and yet must meet the demands of educational policy makers. The factors and guidelines to determine this placement vary greatly between schools and districts. Given that the Unite d States' standing as a scientific and technological leader in the world is greatly dependent on its ability to educate its population and workforce in mathematical ideas, it is critical that teachers and administrators are adequately informed to make such placement decisions. Research is needed to show which factors are reliable and accurate predictors of algebra readiness, to what extent those factors contribute to a student's success and how to best assess these factors. It is necessary to evaluate whi ch practices in student placement have lead to increased student success and which have lead to failure. Teachers' and administrators' understanding of these factors is crucial to raising the math achievement of America's students. Purpose of the Study Th e purpose of this study is to research the predictors and consequences of eighth grade algebra success. A better understanding of the predictors of algebra success will allow schools to more appropriately place students in their eighth and ninth grade mat h courses. The study will use a medium sized suburban school 11 district to analyze student performance on previous assessments of student achievement in math and their corresponding success in algebra. The study will also relate eighth grade algebra achiev ement to college readiness factors such as course rigor and SAT scores among three cohorts of students. Another goal of this study is to gain a working knowledge of current eighth grade math course selection practices in the above mentioned district. With a firm grasp of the current course selection practices and a better understanding of the validity of certain predictors, the study will be able to offer suggestions to this district and others like it as to the best practices for placement in eighth grade algebra. The study will also demonstrate the effects of taking algebra prior to students gaining adequate preparation. With the information that results from this study, educators will be able to make more informed math course selections for students in the eighth grade. More informed math course selection in the eighth grade will allow students to take the course that will most benefit their progression of mathematical skills and thinking. Research Questions The following research questions guided this study: 1. What criteria do intermediate school counselors use to place eighth grade students in their math course? Are students more likely to take advanced math and science courses during high school if they have success in eighth grade algebra? Importance of the Study The findings of this study will give educators and policymakers alike a bett er understanding of the predictors and consequences of eighth grade algebra achievement. While all stakeholders may claim that their decisions are based on the best interests of students, a closer examination of the practices that yield the greatest stude nt achievement will allow educators to make more appropriate decisions on the behalf of students. Educators need to be armed with information that allows students to be placed in the most appropriate math course. It is of paramount importance that all stu dents not only have an opportunity to take and learn algebra but also to develop a working conceptual and procedural knowledge of algebraic principles. Furthermore, basic algebra serves as a foundation upon which a myriad of advanced mathematical skills a re based. Research has shown a clear link between algebra preparedness and algebra success. Schools must be aware of the characteristics of students that are prepared to take algebra and those that are not. By using one of these factors, or a combinatio n of them, schools may be able to better place students in the most appropriate math course, leading to greater math achievement over the course of their academic career. Educators will also be able to determine the reasons that 13 students are not ready to take algebra and address deficiencies in the curriculum and instructional practices. This study will allow schools to best place students in math courses that are best suited to meet their academic needs. The push to enroll students in algebra in the eigh th grade has been driven by the need to see increased math achievement among all students. Policymakers saw that students who took algebra in the eighth grade were more college ready as demonstrated through a more rigorous course load and achieved better SAT scores. However, pushing all students to enroll in eighth grade algebra did not result in increased student achievement. Instead, it placed many students in a course for which they were unprepared. This study will lend support to the notion that inc reased math achievement is not always manifested through more difficult courses at an earlier age. Math achievement increases through a better understanding of and a genuine interest for learning the subject matter. When this study is finished, policymak ers will be able to see that students placed in the appropriate math course for their preparation level show higher math achievement than students who are accelerated into algebra courses prior to being fully prepared. Increased math achievement amongst Am erican students is an appropriate and important goal for the public education system in the United States. There are differing views on what would be the most effective and efficient way to boost this achievement. This study will help to show educators a nd policymakers that eighth grade math course selection is a critical decision that has a direct relation to student's 14 college readiness and give educators suggestions as to the best predictors of algebra success. Assumptions This study assumes that the qu antitative measurements of student success are the result of the students' best efforts to demonstrate their knowledge and mastery of the information to be assessed. It is further assumed that students are currently placed in their eighth grade math cours e based upon seventh grade teacher recommendation. In this district, teacher recommendations are the default course placement criteria and are based on students' performance in their previous math course. The study also assumes that the instruction and c urriculum in Algebra courses follow the California State Standards for Algebra. Limitations This study is set in a high performing, medium sized suburban school district. There is a strong emphasis on college readiness in the homes of the students, at sch ool and in the community. The first limitation is that quantitative results of this study are limited by the small sample size as well as the characteristics of the participants. Certain students may have incomplete academic records for a variety of diffe rent reasons and there is a moderate degree of transiency in the district which makes comparisons between students more difficult. Variables such as ethnicity, socioeconomic status and family background will not be taken into account in this study. 15 The se cond limitation is that student assessments that are used to determine preparedness and student achievement are only snapshots of student learning and may not reflect the true level of the students' knowledge of the California State Standards. Thirdly, the qualitative aspects of this study acquired through interviews and surveys are limited and open to interpretation of the researcher and participants. Thus they may be subject to a certain level of bias. Although measures will be taken to collect data fro m multiple viewpoints, a small level of bias is still a concern of the researcher. This is to be expected in any qualitative study. Full document contains 167 pages Abstract: The nature of Algebra, and the essential fundamentals that children master through studying Algebra serve as an essential building block of further mathematical studies. This study underscores the importance of success in Algebra.
the Algebra Word Problems Tutor and in this class I am very excited about teaching this class because up until now my other courses and in your school work you have learned how to do algebra. You have learned the mechanics of how to do algebra. You have learned how to solve an equation, had a add negative numbers and positive numbers in things like that and then given a problem, you have the tools in your toolbox to try to be able to solve that kind of problem but everybody knows that in the real world and also in your test you are going to be given a certain kind of problem called the word problem and that makes people sometimes have a hard time and the reasons is because your problem is not given to you in a nice one line fashion and it does not just say solve this equation or manipulate this expression. The word problem has a challenge to it because it is bunch of words and use numbers in there and Sally and Jane and different peoples names and so on. And you are challenge really with those kinds of problems because you have to take the information out of the problem and put it on your board or on your paper. And then trying to get to the math usually is not that hard. Trying to get through the math usually is not that difficult the actual solving of the problem. The real challenge with the word problem is actually taking the information out of the problem and constructing to your equations or whatever it is you are trying to do on your paper. That is the real challenge and to be honest with you that is real life. Your real life is not giving you a bunch of one liner homework problems real life is go figure out how to make this airplane fly or go figure out how this engine is running to hot, go figure out why it is running to hot and fix it. So what are we going to do in this course is this course is going to be 100% word problems. That is all it is. The actual math behind it, the actual solving the equation is not going to be that hard and I did that on purpose. The challenge is taking the word problems and taking that information out and putting it into a problem form so you can solve it. That is what we are going do today. So the first section is Algebra word problems involving number. Basically these problems are going to have one number relating to another number sometimes its age, sometime its distance things like that, and I think the word problems just like anything else in life the easiest way to learn how to do them is just to see some examples so let us go and find those examples. The first problem that we are going to do and talk about is on your board here and it says on your screen right here and it says, four times a number increase by two is equal to 30. What is the number? The number one thing in you need to do when you are solving word problems any kind of problem that has a sentence involved in it is called a word problem is you have to understand what is actually asking you. Number one thing people do is you see a word problem and people start trying to write down answers and numbers and equations without actually understanding what the problem is saying and what is asking you. You have to do that so do not try to skip ahead. Read this problem two or three times until you understand what is actually asking you to do then start writing down your equations and your numbers and so on. So let us read it one more time. Four times a number increase by two is equal to 30. What is that number? So we have a number that is what we are trying to find and we multiply it times four and then we take that thing and then we add two to it and then the resulting answers should be equal to 30 and what we are trying to do is try to find that number. So what we are going to do because we have an unknown here, we have a number okay we are going to say that the variable x is equal to the number. The reason I am doing this is because I really do not know what the number is that is the whole point, I do not know what the number is so I am going to assign a variable to it. And remember way back to being in algebra that is what are you doing here you do not know what it is. So it is a variable, it is the unknown. So you assign it to be x. So what you want to do is you want to translate this problem into an equation or something you can solve. So four times a number increased by two. Well if my number is what I am trying to solve for, then four times a number must be 4x four times a number. Increase by two then I am going to add two to it. It has got to be equal to 30. Congratulations you have just translated your first word problem into an equation. Four times a number increase by two is equal to 30 and now what we have to do is solve for our unknown. So to do that, going back to your basic algebra we will subtract two from both sides so this two will go away two minus two is zero and on the right side we will have 30 minus two. So all have I done is to subtract two from the left and get rid of that two and two from the right, I actually have not done any math yet so 4x is equal to 28 and then into it you solve for x. You divide both sides by four so x is going to equal to 28 divided by four so on the left 4x divided by four will give you one so that is why you have one x there and 28 divided by four. So x is going to be equal to seven. X is going to be equal to seven. So you see what you did here is you had a word problem that you translated into an equation 4x+2=30 and you just use your math to solve that equation and here is your answer. Now the great thing about word problem, actually the great thing about algebra in general but especially word problems is a you know you can always check your work always 100% almost always you can check your work and you should do that with every problem and it is really easy to do because now you that figured out what number is which we said in now it is equal to seven you want to see if that is the right answer. So that is the great thing when you are taking a test you will always know if it is right. So do not pass up the opportunity to figure out if you did the right thing. So now if your number is equal to seven which is what we said let us just read the problem one more time. Four times a number four times a number see now we figured out what this number is let us put in here four times a number increased by two, this has got to be equal to 30. Well, let us see what it is equal to seven times four is 28 plus two; we will just keep that there because we have not done anything yet. 28 plus two is equal to 30 okay. We have done the right thing. We know we have done the right thing because we done our math, we found our number, and then when we apply the logic back to our number that we found, we get exactly what we should get. The second problem is a little bit more complicated but not too much more complicated we are just going to build in complexity here. Second problem the sum of two numbers is 30 and one number is two times the other number. What are the two numbers or find the two numbers so again same thing as before you have to read the problem, understand the problem, and then start writing stuff on the paper, okay. The sum of two numbers now remember the word sum, the word sum is adding two things together. The sum of two numbers is 30 and one number is two times the other number. Find the numbers. So let us go ahead and just tackle this one step at time. Now we have two numbers. We all have one number, we have two numbers and you can tell because the problem might there says we have two numbers so you know this so we can say that a variable x is equal to number one and the variable y is equal to number two. Now in this problem we know we have to have two numbers because it says right here in the very first sentence the sum of two numbers is 30. So you know and you are trying to find the two numbers you know you are going to have two variables and that what tells you, you want to have two variables right there. So let us go ahead and translate that first sentence. The sum of the two numbers x+y that is the sum of these two numbers that we are trying to find is 30. So we have already translate
Puzzler algebra lesson helps students explore polynomials by solving puzzles. The activity explains the relationship between expanding and factoring polynomials, as well as factoring trinomials, and multiplying monomials and binomials. The lesson includes an activity sheet, downloadable in PDF format. The material is appropriate for grades 9-12 and should require 1 class period to complete.Fri, 31Mathematics and Student Life Skills course, designed for Miami Dade Community College, integrates arithmetic and beginning algebra for the undergraduate student. By applying math to real-life situations most students experience during college, the instructors attempt to make math both fun and applicable. The instructors specifically wish to dissipate the anxiety many college students feel when approaching math at an advanced level. Students can use the information provided on this website to help apply mathematical concepts to their own lives, while instructors can use the assignments, syllabus, and lecture notes to create their own relevant assignments in a mathematics course.Mon, 5 May 2008 03:00:13 -0500BasketMath Interactive by Science Academy Software, this site is a collection of math questions on subjects including basic arithmetic, order of operations, calculating perimeters and distance, exponents, and bar graphs. It is an excellent and fun math review for students of all ages, and a ready-to-use classroom exercise for educators.Wed, 9 Apr 2008 03:00:03 -05Algebra Through Problem Solving non-traditional Algebra text (high school and early college levels) placed on the Web by the Science Education Team at Los Alamos National Laboratory. Browse it on the Web or download a PDF version. Chapter headings include: The Pascal Triangle; The Fibonacci and Lucas Numbers; Factorials; Arithmetic and Geometric Progressions; Mathematical Induction; The Binomial Theorem; Combinations and Permutations; Polynomial Equations; Determinants; and Inequalities. Problems (and answers to the odd-numbered problems) are provided for each section.Mon, 6 Aug 2007 03:00:02 -0500xyAlgebra students in college-level math courses, revisiting algebra can be a challenge, and learning for it the first time can prove to be tricky as well. This website, created by Professor John Miller of the City College of the City University of New York (CUNY) contains a very helpful tutorial application that can help students master different aspects of algebra. This application features explanations of 320 different aspects of algebra, step-by-step sample problems, and even hints to help students along their way. Visitors will also appreciate the fact that the website contains several demos and a set of resources for instructors.Fri, 20 Jul 2007 03:00:02 -0500Professor Freedman's Math Help Power is a site created by Professor Freedman, a highly acclaimed teacher of basic mathematics. She has several resources to help students of all ages learn pre-algebra and elementary algebra skills. Many sample homework assignments are provided, as well as general study tips and a list of "ten ways to reduce math anxiety." A collection of video clips demonstrate specific math concepts. There are also Java games that are a diversion from straight math, and most are educational. Some sections of the site are intended for mathematics instructors, like the Ten Commandments for Math Teachers.Tue, 6 Mar 2007 03:00:01 -0600algebasics the motto, "Show me how, now!" algebasics is a fine online mathematics instructional resource that takes young and old alike through the basics of algebra. The breadth of the material is divided into sixteen sections, which begin with, appropriately, "the basics", and proceed all the way to a section on applying algebra to real-world situations. Each section asks users to solve a number of problems so that they will gain mastery of each concept. The interface deployed here is quite user-friendly, as each problem is narrated so that users will better understand the process needed to complete each problem successfully. Overall, it's a well-designed introduction to this area of the mathematical universe, and one that is very easy to use.Sun, 4 Feb 2007 03:00:05 -0600
When the numbers just don't add up... Following in the footsteps of the successful The Humongous Books of Calculus Problems , bestselling author Michael Kelley has taken a typical algebra workbook, and made notes in the margins, adding missing steps and simplifying concepts and solutions. Students will learn how to interpret and solve problems... more... "Integers" is a refereed online journal devoted to research in the area of combinatorial number theory. It publishes original research articles in combinatorics and number theory. This work presents all papers of the 2013 volume in book form. In addition,it containsan introductory survey paper. more... The notion of closure pervades mathematics, especially in the fields of topology and projective geometry. Demonstrating this pervasiveness in the field, this graduate-level book provides a complete introduction to closure systems. With an emphasis on finite spaces and algebraic closures, the text covers graph theory, ordered sets, lattices, projective... more... Linear Algebra and Matrix Analysis for Statistics offers a gradual exposition to linear algebra without sacrificing the rigor of the subject. It presents both the vector space approach and the canonical forms in matrix theory. The book is as self-contained as possible, assuming no prior knowledge of linear algebra. The authors first address... more... Based on presentations given at the NordForsk Network Closing Conference ?Operator Algebra and Dynamics,? held in Gjáargarður, Faroe Islands, in May 2012, this book features high quality research contributions and review articles by researchers associated with the NordForsk network and leading experts that explore the fundamental role of operator algebras... more... Algebra: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using Algebra. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all the key areas of algebra including elementary operations, linear equations,... more...
This electronic journal, published by the European Association for Programming Languages and Systems, features "articles ranging from the theoretical foundations of functional and logic programming up to the application... The Journal of Graph Algorithms and Applications found at this Web site is the electronic version of the scientific journal with the same name. It is a collection of research papers dealing with the "analysis, design,... Maths Challenge is a fun activity to build mathematical and reasoning skills. Each month during the school year, a new set of math problems are posted online. There are miscellaneous problems at junior and senior levels... For students in college-level math courses, revisiting algebra can be a challenge, and learning for it the first time can prove to be tricky as well. This website, created by Professor John Miller of the City College of... A computer science professor at the State University of New York maintains this online repository, which serves "as a comprehensive collection of algorithm implementations for over seventy of the most fundamental...
Geometer's Sketchpad The Geometer's Sketchpad is a dynamic construction, demonstration, and exploration tool that adds a powerful dimension to the study of mathematics. You and your students can use this software program to build and investigate mathematical models, objects, figures, diagrams, and graphs. With Sketchpad™, you can give your students a tangible, visual way to explore and understand core concepts—from numbers and operations, algebraic thinking, and geometry and measurement in elementary and middle school to algebra, geometry, trigonometry, precalculus, and calculus in high school and college. Concepts that students frequently find difficult become very clear when they see visual representations on the screen and interact with them using Sketchpad. Geometer's Sketchpad allows students to visually represent math concepts and ideas in powerful ways and supports the notion of multiple means of representation Geometer's Sketchpad allows students to demonstrate their knowledge using the computer ans supports the notion of multiple means of expression
 Geometry  Deciding how much paint or carpeting to buy.  Deciding how much fencing you need to enclose your yard.  Deciding if and how your new furniture will fit into your room. Probability and Statistics  Understanding health risks.  Understanding and making business decisions.  Calculus and Differential Equations  Needed for engineering, business, economics, physics, astronomy.  Used for problems that involve work, water pressure, etc. Sciences  Help in understanding yourself and the world you live in. Facts about college math andscience courses:  Assignments are crucial.  Class time may focus more on theories and ideas than on problem- solving.  Class size may be larger.  Knowledge of technology may be important. Effective Use of Math and ScienceTextbooks Involves:  Recognizing that material is presented sequentially.  Taking notes, working examples yourself, identifying things you don't understand.  Paying attention to formulas.  Using memory techniques, especially in learning new vocabulary.  Learning the language of the discipline.  Keeping up with reading and homework. Studying and Homework  Review materials regularly and often.  Do problems, problems, more problems.  Don't let frustration stop you.  Work with a partner or study group.  Use what you know about your learning styles. Steps in Solving Word Problems 1. Read the problem carefully and understand what it is asking. 2. Relate to problems you have done and identify possible solutions. 3. Pick a way to solve the problem and do it. 4. Review your results. PROBLEM-SOLVINGSTRATEGIES  Look for a pattern.  Make a table.  Break into smaller parts.  Check a similar problem.  Work backwards.  Draw a table/diagram.  Translate words into an equation. Math and Science Test-Taking Tips  Read through the entire exam first.  Analyze problems carefully.  Estimate an approximate answer before working the problem.  Break solution into smallest possible steps.  Draw a picture to help you see the problem.  Be neat.  Use the opposite operation to check your work.  Look back at the question to be sure you followed directions. Visual Aids  Provide a lot of information in an easy to understand form in a relatively small amount of space.  Show trends over time.  Show relative rankings.  Show distributions.  Show cycles.
Complex numbers as ordered pairs of reals, Representation of complexnumbers in the form a+ib and their representation in a plane, Argand diagram,algebra of complex numbers, modulus and argument (or amplitude) of acomplex number, square root of a complex number, triangle inequality,Quadratic equations in real and complex number system and their solutions.Relation between roots and co-efficients, nature of roots, formation of quadraticequations with given roots. UNIT 3 : MATRICES AND DETERMINANTS: Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three.Properties of determinants, evaluation of determinants, area of triangles using determinants. Adjointand evaluation of inverse of a square matrix using determinants and elementary transformations,Test of consistency and solution of simultaneous linear equations in two or three variables usingdeterminants and matrices. UNIT 4 : PERMUTATIONS AND COMBINATIONS: Fundamental principle of counting, permutation as an arrangement andcombination as selection, Meaning of P (n,r) and C (n,r), simple applications. UNIT 5 : MATHEMATICAL INDUCTION: Principle of Mathematical Induction and its simple applications. UNIT 6 : BINOMIAL THEOREM AND ITS SIMPLE APPLICATIONS: Binomial theorem for a positive integral index, general term and middle term,properties of Binomial coefficients and simple applications. Ordinary differential equations, their order and degree. Formation of differentialequations. Solution of differential equations by the method of separation ofvariables, solution of homogeneous and linear differential equations of the type: dy+ p (x) y = q (x)dx UNIT 11: CO-ORDINATE GEOMETRY: Cartesian system of rectangular co-ordinates 10 in a plane, distance formula,section formula, locus and its equation, translation of axes, slope of a line,parallel and perpendicular lines, intercepts of a line on the coordinate axes.Straight linesVarious forms of equations of a line, intersection of lines, angles betweentwo lines, conditions for concurrence of three lines, distance of a point from aline, equations of internal and external bisectors of angles between two lines,coordinates of centroid, orthocentre and circumcentre of a triangle, equation offamily of lines passing through the point of intersection of two lines.Circles, conic sections Standard form of equation of a circle, general form of the equation of a circle, its radius and centre,equation of a circle when the end points of a diameter are given, points of intersection of a line and acircle with the centre at the origin and condition for a line to be tangent to a circle, equation of thetangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) instandard forms, condition for y = mx + c to be a tangent and point (s) of tangency. UNIT 12: THREE DIMENSIONAL GEOMETRY: Coordinates of a point in space, distance between two points, section formula,direction ratios and direction cosines, angle between two intersecting lines.Skew lines, the shortest distance between them and its equation. Equations of aline and a plane in different forms, intersection of a line and a plane, coplanarlines. UNIT 13: VECTOR ALGEBRA: Vectors and scalars, addition of vectors, components of a vector in twodimensions and three dimensional space, scalar and vector products, scalarand vector triple product. PHYSICSThe syllabus contains two Sections - A and B. Section - A pertains to the Theory Part having 80%weightage, while Section - B contains Practical Component (Experimental Skills) having 20%weightage.SECTION – A Force and Inertia, Newton's First Law of motion; Momentum, Newton's SecondLaw of motion; Impulse; Newton's Third Law of motion. Law of conservation oflinear momentum and its applications, Equilibrium of concurrent forces. Static and Kinetic friction, laws of friction, rolling friction. Dynamics of uniform circular motion: Centripetal force and its applications. UNIT 4: WORK, ENERGY AND POWER Work done by a constant force and a variable force; kinetic and potentialenergies, workenergy theorem, power. Potential energy of a spring, conservation of mechanical energy, conservativeand nonconservative forces; Elastic and inelastic collisions in one and twodimensions. UNIT 5: ROTATIONAL MOTION Centre of mass of a two-particle system, Centre of mass of a rigid body; Basic concepts of rotationalmotion; moment of a force, torque, angular momentum, conservation of angular momentum and itsapplications; moment of inertia, radius of gyration. Values of moments of inertia for simplegeometrical objects, parallel and perpendicular axes theorems and their applications. Rigid bodyrotation, equations of rotational motion. UNIT 6: GRAVITATION The universal law of gravitation. Acceleration due to gravity and its variationwith altitude and depth. Kepler's laws of planetary motion. Gravitationalpotential energy; gravitational potential. Escape velocity. Orbital velocity of asatellite. Geo-stationary satellites. UNIT 7: PROPERTIES OF SOLIDS AND LIQUIDS Elastic behaviour, Stress-strain relationship, Hooke's Law, Young's modulus, bulkmodulus, modulus of rigidity. Pressure due to a fluid column; Pascal's law and its applications.
*Abstract Algebra: Intro - 3rd edition Summary: ABSTRACT ALGEBRA: AN INTRODUCTION is intended for a first undergraduate course in modern abstract algebra. The flexible design of the text makes it suitable for courses of various lengths and different levels of mathematical sophistication, ranging from a traditional abstract algebra course to one with a more applied flavor. The emphasis is on clarity of exposition. The thematic development and organizational overview is what sets this book apart. The chapters are organized around th...show moreree themes: arithmetic, congruence, and abstract structures. Each Them is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another. ...show less Almost new condition. SKU:978111156962482126.58 +$3.99 s/h New Textbookcenter.com Columbia, MO Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book $140
I think linear transformations are abstract by nature. Of course you can construct geometrical analogies in many cases, like for projection operators, rotations etc, and you might be able to use such examples to guide students towards the general definition. i used to try all kinds of examples of linear phenomena. E.g. cooking recipes. Doubling the ingredients of the recipe doubles the output. the main job is to convey the idea of linearity, outputs that change proportionately to the inputs. then a linear transformation is any operation that behaves like this. e.g. differentiation behaves linearly on functions. projections of one space onto a lower dimensional space are linear. but force is not linear with speed, i.e. F = MA, so force is proportional to acceleration. genericusrnme #4 Mar5-12, 07:40 AM P: 615 Linear Transformations in Linear algebra I'd introduce them by showing what they actually do, take you from one space to another. The way Gilbert Strang does it on the MIT opencourseware linear algebra course is pretty good if you want to get introduced to what they do imo mathwonk #5 Mar7-12, 12:25 PM Sci Advisor HW Helper P: 9,421 Since a primary application is to differential equations, with students who have had calculus it seems important to point out that differentiation is linear. When acting on polynomials of fixed degree it also gives the basic example of a nilpotent linear operator, not an intuitive idea without that example. And when acting on spaces of exponential functions it gives the fundamental example of eigenvectors and eigenvalues, another absolutely crucial concept to acquire. DrummingAtom #6 Mar10-12, 06:40 PM P: 660 I learned linear algebra best when I thought in terms of geometry. Unfortunately, linear algebra starts in Rn from the start which is pretty annoying from someone like me. I made everything into a simpler case in R2 or R3. Without writing my own thoughts I found a good link for how I would best learn this. The price is slowly going up because the editions are getting farther along. I have the 2nd edition and it's wonderful for showing the intuitive and visual representation of linear algebra. This is how math should be taught.. at least for learners like me.
Precalculus: Mathematics for Calculus 9780840068071 ISBN: 0840068077 Edition: 6 Pub Date: 2011 Publisher: Brooks Cole Summary: Designed to give students a background in mathematics theory and introduce them to mathematics concepts this textbook is comprehensive without being daunting. Students are introduced to modelling and problem solving and they are given a rigorous workout on what they have learned as they work through the book. It has many graphs that chart mathematical ideas that students can assimilate with ease. It is written in a c...lear and readable style that will aid comprehension and enjoyment. This is just one of the many cheap math textbooks we have available for students to acquire in great condition. Stewart, James is the author of Precalculus: Mathematics for Calculus, published 2011 under ISBN 9780840068071 and 0840068077. Six hundred thirty six Precalculus: Mathematics for Calculus textbooks are available for sale on ValoreBooks.com, two hundred seventeen used from the cheapest price of $87.21, or buy new starting at $205.85 International Edition: Almost new condition. SKU:9780840068866-2-0-15 Orders ship the same or nex... [more]6 edition, , color Printing, Softcover International edition **Different ISBN and Cover design but offer same content and paper quality as that of domestic US editi [more] The book itself was least useful to me the only reason I had to buy the book was to do online tests with the access code. These codes are unfair to the student, because they make getting your money back on books that are not even used except for one code nearly impossible. Precalculus, required that I "use" this book even though the book was never actually used only one page was ever opened in this book, and that page was to get the UNFAIR online code. This book did a very good job of explaining concepts step by step. Compared to other math textbooks, this one was actually easy to follow, the problems and examples were similar and flowed in a logical way. I learned about vectors, conic equations,trigonometry and various functions from this book.
This is a free, online wikibook, so its contents are continually being updated and refined. According to the authors, "When... see more This is a free, online wikibook, so its contents are continually being updated and refined. According to the authors, "When using numerical methods or algorithms and computing with finite precision, errors of approximation or rounding and truncation are introduced. It is important to have a notion of their nature and their order. A newly developed method is worthless without an error analysis. Neither does it make sense to use methods which introduce errors with magnitudes larger than the effects to be measured or simulated. On the other hand, using a method with very high accuracy might be computationally too expensive to justify the gain in accuracy.״ ״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a... see more ״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a surprisingly deep subject. We assume that students have some familiarity with basic set theory, and calculus. But very little of this nature will be needed. To a great extent the book is self-contained. It requires only a certain amount of mathematical maturity. And, hopefully, the student's level of mathematical maturity will increase as the course progresses. Before the course is over students will be introduced to the symbolic programming language Maple which is an excellent tool for exploring number theoretic questions.״ ״This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties,... see more ״This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text.״ ״Numerical Computing with MATLAB is a textbook for an introductory course in numerical methods, MATLAB, and technical... see more ״Numerical Computing with MATLAB is a textbook for an introductory course in numerical methods, MATLAB, and technical computing. It emphasizes the informed use of mathematical software. Topics include matrix computation, interpolation and zero finding, differential equations, random numbers, and Fourier analysis. Based on MATLAB, the textbook provides more than 70 M-files. Many of the more than 200 exercises involve modifying and extending these programs. The book also makes extensive use of computer graphics, including interactive graphical expositions of numerical algorithms." Individual chapters can be downloaded as separate pdf files and while curriculum tools and materials can also be downloaded. This is a free, online textbook "is written primarily for engineering and science undergraduates taking a course in Numerical... see more This is a free, online textbook "is written primarily for engineering and science undergraduates taking a course in Numerical Methods. The textbook offers a unique treatise to numerical methods which is based on a holistic approach and short chapters.״ In addition to the text, one can also access Video Lectures, PowerPoint Presentations,Worksheets, Multiple-Choice Tests, and Anecdotes.
McGraw-Hill's GED MathematicsAbout the Book Problem-solving and computational skills, with special focus on the use of the Casio FX-260 calculator, understanding grids, and strategies for handling word problems
Browse Results Modify Your Results The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and instructors. This first volume focuses on the analysis of real-valued functions of a real variable. Besides developing the basic theory it describes many applications, including a chapter on Fourier series. It also includes a Prologue in which the author introduces the axioms of set theory and uses them to construct the real number system. Volume II goes on to consider metric and topological spaces and functions of several variables. Volume III covers complex analysis and the theory of measure and integration
Dunstable PrecalculusAlthough elements of continuity do exist (emphasis on ratios is Geometry is very algebra intensive, and the first introduction to trigonometry often starts in geometry), Geometry stands apart from the other traditional math courses with its emphasis on visualization and formal proofs, and the pro... ...One of the prerequisite courses of this course is linear algebra. In general to join a PhD program a student should pass at least three branches of math namely, Analysis, Geometry and Algebra. Analysis include real and complex analysis (advanced calculus is part of analysis) Algebra include abs
Modern Geometries - 4th edition Summary: This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. This edition reflects the recommendations of the C...show moreOMAP proceedings on Geometry's Future, the NCTM standards, and the Professional Standards for Teaching Mathematics. References to a new companion text, Active Geometry by David A. Thomas encourage students to explore the geometry of motion through the use of computer software. Using Active Geometry at the beginning of various sections allows professors to give students a somewhat more intuitive introduction using current technology before moving on to more abstract concepts and theorems
pages: 10 size: 397.00 KB You might think you need calculus to determine the area between the tire tracks made by this bike, ridden by Jason McIlhaney, BS 2000. Surprisingly, geometry of ... pages: 3 size: 165.00 KB Practice Problems for Math 131 Test 1 THESE PROBLEMS ARE FOR PRACTICE. THEY ARE NOT MEANT TO REPLACE THE RECOMMENDED HOMEWORK PROBLEMS, AND MAY OR MAY NOT REPRE ... pages: 6 size: 299.00 KB Practice Problems for Math 131 Test 1 THESE PROBLEMS ARE FOR PRACTICE. THEY ARE NOT MEANT TO REPLACE THE RECOMMENDED HOMEWORK PROBLEMS, AND MAY OR MAY N ... pages: 4 size: 90.00 KB Algebra II Review Exponential Functions and Logarithms Answer Key*** Although there is not a specific SPI for this activity, there are two checks for un ... pages: 13 size: 127.00 KB 121Section 4.1 - Exponential FunctionsExponential functions are extremely important in both economics and science. It allows us to discuss the growth ...
NOTES - Lawton Chiles Middle School Covers most topics in a pre-algebra curriculum Algebra With Pizzazz! in a Binder Covers most topics in a first-year algebra curriculum . What Sound Do Two Porcupines like terms.pdf 32.17 kB000 stemteachersnowpdproject.wikispaces.com A major focus of pre-algebra courses is the familiarize students with variable ... Simplify expressions to decode answer to a riddle (see Algebra with Pizzazz ... lesson order of operations.docx
Summary: James meticulou...show mores accuracy, patient explanations, and carefully graded problems. The content has been refined and the examples and exercises have been updated. In CALCULUS: CONCEPTS AND CONTEXTS, this well respected author emphasizes conceptual understanding - motivating students with real world applications and stressing the Rule of Four in numerical, visual, algebraic, and verbal interpretations. All concepts are presented in the classic Stewart style: with simplicity, character, and attention to detail. In addition to his clear exposition, Stewart also creates well thought-out problems and exercises. The definitions are precise and the problems create an ideal balance between conceptual understanding and algebraic skills. ...show less James meticulous accuracy, patient explanations, and carefully graded problems. Other Editions of Calculus : Concepts and Contexts Single Variable -Text Only: Shows some signs of wear, and may have some markings on the inside. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy! $20.95 +$3.99 s/h Good LotsofBooks Nashville, TN No comments from the seller $125.00 +$3.99 s/h New bluehouse acton, MA Brand new. $179179
The Department of Mathematical Sciences seeks to provide students with the training in mathematics and statistics necessary for success in their careers. The instructional mission of the department can be divided into three categories: 1) we offer quantitative literacy to general university students; 2) we provide the mathematical background for students preparing for careers in other fields; and 3) we offer a quality program of mathematical specialization in the form of majors (with various options) and a minor. Graduates of major/minor programs may teach in public schools or work in national laboratories, government or businesses. Our graduates often seek advanced degrees in mathematics. The University Catalog defines the mathematical literacy that is expected of all students at The University of Montana. A graduate will demonstrate the ability to a) formulate real-world problems quantitatively, b) solve quantitative problems, c) interpret solutions to problems, and d) make critical judgments regarding the validity of competing formulations and solutions. This definition is considered in the established learning goals of all undergraduate courses. There is the commitment to maintain a faculty make-up with adequate expertise to carry out its service obligations to the University of Montana through course offerings. Naturally, the research commitment of the department is more extensive at the graduate level, but undergraduate students often participate through research courses, seminars, independent study, undergraduate projects, and theses. Steps in the Assessment Process 1. Student Learning Goals The following are the most important learning goals for students who complete a major or minor in mathematical sciences: Students are expected to obtain a broad background in mathematics as demonstrated by their understanding of core mathematical content: functions, calculus, linear algebra, proof. Students will develop the ability to communicate mathematics both within and outside the discipline. Students will develop clear, analytical thinking skills as demonstrated by rigorous reasoning in mathematical arguments. Students will employ a variety of problem solving strategies, including the use of technology. Students will be prepared for advanced, more specialized study in mathematics (pure and applied), statistics, or mathematics education. 2. Measurement of Goals The following means are used to measure the progress of our students and to evaluate the programs that we offer: For each undergraduate class, we have identified specific learning goals. Expected entering skills and prerequisites are considered. Suggestions are given for in-class measurement of the learning goals. Potential problems (for example, future needs of students and difficulties encountered when multiple sections converge into one subsequent course) are part of the evaluation. Because the content of previous mathematics courses is extremely relevant to future study in the field, we constantly monitor the success of students in subsequent courses. The success of students in the next course is a good measure of their command of the previous courses. We follow up on our graduates. We try to stay in touch to see if they think we have prepared them well. We often receive feedback from our graduates who continue in academic programs regarding the value of their undergraduate preparation here. Many of our students participate in independent projects leading to publications and public presentations. Mathematics students may also participate in local, regional, and national competitions. 3. Modifications Based on Assessment The following is a sample of changes made after assessing program offerings in the department: Following a retreat of the entire faculty to study our calculus offerings, a committee undertook a year-long restructuring of the course. All aspects were considered, including prerequisites, the use of technology in instruction, the future goals of students in the courses, consistency in multiple sections, and grading. In conjunction with the Provost's Quality Initiative, a Tutorial Center was established to benefit students in entry-level courses. The Center employs advanced mathematics students to work with students in 100-level courses. All mathematics courses are evaluated for their technology needs. Computer labs and mathematical software are constantly monitored and upgraded to ensure that we provide the best technology support possible. Laboratory courses have been added to supplement some course offerings (Math 317, Math 388, and Math 418) and technology components have been expanded internally in some classes (Math 150, Math 221, and Math 431). Advanced statistics labs (Math 447 and Math 448) are now taught by professors, not adjunct faculty. Program adjustments have been proposed, debated, and implemented. For example, in our Combinatorics and Optimization emphasis, Math 481 and Math 482 were eliminated and Math 485 Graph Theory was introduced. Regarding mathematics literacy, the department experimented with large lecture classes in four courses (Math 107, Math 117, Math 121, and Math 150). The results indicated that students were adversely affected by this format, so we returned to self-contained sections in Math 107 and Math 121. The hiring of Lecturers whose primary responsibility is teaching of entry-level courses has led to a more stable cohort of faculty. In addition, the department has committed to having more tenure-track faculty teaching and coordinating lower-level classes.
Saugus StatisticsStudents will benefit most from reading and interpreting highly regarded scientific journalism, introductory texts, and essays about scientific topics. Mathematics is the exploration of how quantities relate to each other. Students usually find trouble in mathematics when they lose sight of its ultimate simplicity
The Geometer's Sketchpad a scope that spans the mathematics curriculum from middle school to college, The Geometer's Sketchpad brings a powerful dimension to the study of mathematics. Sketchpad is a dynamic construction and exploration tool that enables students to explore and understand mathematics in ways that are simply not possible with traditional tools or with other mathematics software programs. What's new in this version: Bug Fixes: Functions defined by a Drawing no longer fail to fit the greatest height of the Drawing when the Drawing is highly rotated. Custom Tools that both match certain Givens to existing sketch objects and create points-on-objects on their Givens no longer occasionally fail to match properly. Several example documents in the "Elementary and Middle School Mathematics" folder have undergone minor improvements. (Macintosh) Sketchpad is no longer incompatible with OS X Paren
More About This Textbook Overview Number Power 4: Geometry introduces students to working with lines, angles, triangles and other plane figures, and solid figures. Step-by-step instruction and extensive reinforcement exercises ensure that students are solidly grounded in the basics of geometry
Precalculus Quick Links Webinars Unit Downloads Exponential and Logarithmic Functions The two simplest families of transcendental functions are those of exponential functions (of the form y = bx) and their inverses - the logarithmic functions. The growth behavior of exponentials are compared with power functions and the use of exponential and logarithmic transformations on data are treated, as well as an application to compound interest. This lesson involves three real-world data sets in which the relationship between each pair of variables is non-linear. Students will be asked to describe the original relationship between each pair of variables, and observe how each transformation is used to achieve a linear relationship.
Subject: Mathematics (9 - 12) Title: Burst the Inverse! Description: This lesson allows students to investigate functions and their inverses by hand and using graphing calculators. Students will also use equations and graphs. Students will work cooperatively to develop and justify ideas/conjectures about functions and inverses. This is a College- and Career-Ready Standards showcase lesson plan.
Results for ashford university mat 126 survey of mathematics book Sponsored High Speed Downloads ashford university mat 126 survey of mathematics book [Full Version] Essays on AshfordMat 126 WeekThree Written Assignment for students to reference for free. Use our essays to help you with your writing 1 - 23. Question 1 A coin is tossed and then a die is rolled. Find the probability of getting a 5 on the die given that the coin landed tails up. Question 2 Express 3.46 as a ... Suppose the production function for high-quality Brandy is given by: q=K^1/2L^1/2, where q is the output of brandy per week and L is labor hours per week. Why Course Hero? Used Course Hero for the first time yesterday and I cannot wait to use it again. I was extremely confused on my accounting test and how to read the ... AshfordMAT 126: ASurvey of Mathematical Methods Week 4 ... Question 1 A coin is tossed and then a die is rolled. Find the probability of getting a 5 on the die given ... An authoritative introduction to implementing DotNetNuke Web sites, by experienced DotNetNuke implementers and trainersAn impressive author team shows you how to ...
Summary: This text is for a one-term course in intermediate algebra, for students who have had a previous elementary algebra course. A five- step problem-solving process is introduced, and interesting applications are used to motivate students. Coverage progresses from graphs, functions, and linear equations to sequences, series, and the binomial theorem. New to this edition are sections on connecting concepts, study tips, and exercises designed to foster intuitive problem so...show morelving. Bittinger teaches at Indiana University; Ellenbogen at Community College of Vermont. ...show less Hardcover Fine 0321557182 Like New copy, without any marks or highlights. has shelf wear on covers. This is Student US Edition. Same day shipping with free tracking number. Expedited shipping avail...show moreable. A+ Customer Service! ...show less $46.01 +$3.99 s/h LikeNew AZBOOKA Oceanside, CA 0321557182 Sealed CD included! Please refer to the Condition Guidelines and see our thousands of Customers feedback for assurance. $57.75 +$3.99 s/h Good invisibledog Salt Lake City, UT 0321557182 Unmarked text. CD-ROM still sealed. $115.00 +$3.99 s/h New bluehouse acton, MA Brand new. $121.24 +$3.99 s/h New Lyric Vibes Geneva, IL Hardcover New 0321557182 New Condition ~~~ Right off the Shelf-BUY NOW & INCREASE IN KNOWLEDGE... $171.03 +$3.99 s/h LikeNew Bookfriendz Tucson, AZ Hardcover Fine 0321557182 Includes unopened CD! Outstanding Condition. Buy with confidence....Customer Service is our TOP PRIORITY! Delivery confirmation
Tag Archives: theorems When math books present a theorem they almost always immediately present its proof. I tend to skip the proofs. I assume they are correct. I want to get on with the ideas in the text. Later I may come back and read through them. Is this a good thing to advise students to do? I don't know. There are reasons to read proofs. One reason is to help understand why a theorem is true, by seeing the reasoning that lead to the result. Another is to check the reasoning yourself. A third is to learn how to do proofs. Learning to write original proofs is not usually one of the goals of a beginning calculus course. That comes later in a course with "analysis" in its title. There are many theorems that involve some one-off that rarely will be used again. I'm thinking of a proof like that of the sum of the limits is equal to the limit of the sums, where you add and subtract the same expression and this more complicated form allows you to group and factor the terms of the numerator and arrive at the result. Another example is in the Mean Value Theorem where you consider a new function that gives the vertical distance between a function and its secant line. These always bring the question, "How did you know to do that?" If a student can accept things like that, then the proof is usually easy enough to follow. But I would never spend a lot of time making every student fight his or her way through each and every proof. On this other hand, I would never just present a theorem and not give some explanation as to why it is true (and why it is important enough to mention). Unfortunately, I have seen teachers write the Fundamental Theorem of Calculus on the board and proceed to show how to use it to evaluate definite integrals, with no hint of why this important theorem is true. Sure kids can memorize it and use it, but it seems to me they should also have a hint as to why it is true. Some theorems are easy to understand if explained in ways other than a proof. For an example of this, see my post of October 1, 2012 on the Mean Value Theorem. Almost every book will bail out on the Intermediate Value Theorem by claiming (quite rightly) that, "the proof is beyond the scope of this book," or they give the proof in an appendix. But a simple drawing will convince you that it is true. So my feeling is that you do not need to labor over a proof for every theorem, BUT, big BUT, you should provide a good explanation of why it is true. This is important for all students and especially for young women. Jo Boaler writes "As I interviewed more and more boys and girls, I noticed that the desire to know why was something that separated the girls from the boys. The girls were able to accept the method that were shown them and practice them, but they wanted to know why they worked, where they came from, and how they connected with other methods…. When they could not get access to the depth of understanding they wanted, the girls started to turn away from the subject…. Classes in which students discuss concepts, giving them access to a deep and connected understanding of math, are good for boys and girls. Boys may be willing to work in isolation on abstract rules, but such approaches do not give many students, girls or boys, access to the understanding they need. In addition, high-level work in mathematics, science and engineering is not about isolated, abstract rule following, but about collaboration and connection making." Like this: Continuing with some thoughts on helping students read math books, we will now look at the main things we find in them in addition to definitions which we discussed previously: theorems and axioms. An implication is a sentence in the form IF (one or more things are true), THEN (something else is true). The IF part gives a list of requirements, so to speak, and when the requirements are all met we can be sure the THEN part is true. The fancy name for the IF part is hypothesis; the THEN part is called the conclusion. Implications are sometimes referred to as conditional statements – the conclusion is true based on the conditions in the hypothesis. An example from calculus: If a function is differentiable at a point, then it is continuous at that point. The hypothesis is "a function is differentiable at a point", the conclusion is "the function is continuous at that point." This is often shortened to, "Differentiability implies continuity." Many implications are shortened to make them easier to remember or just to make the English flow better. When students get a new idea in a shortened form, they should be sure to restate it so that the IF part and the THEN part are clear to them. Don't let them skip this. Related to any implication are three other implications. The 4 related implications are: The original implication: if p, then q. The converse is formed by interchanging the hypothesis and the conclusion of the original implication: if q, then p. Even if the implication is true, the converse may be either true or false. For example the converse of the example above, if a function is continuous then it is differentiable, is false. The inverse is formed by negating both the hypothesis and the conclusion: if p is false, then q is false. For our example: if a function is not differentiable, then it is not continuous. As with the converse, the inverse may be either true or false. The example is false. Finally, the contrapositive is formed by negating both the original hypothesis and conclusion and interchanging them, if q is false, then p is false. For our example the contrapositive is "If a function is not continuous at a point, then it is it is not differentiable there." This is true, and it turns out a useful. One of the quickest ways of determining that a function is not differentiable is to show that it is not continuous. Another example is a theorem that say if an infinite series, an, converges, then . This is most often used in the contrapositive form when we find a series for which ; we immediately know that it does not converge (called the nth-term test for divergence). The original statement and its contrapositive are both true or both false. Likewise, the converse and the inverse are both true or both false. Any of the 4 types of statements could be taken as the original and the others renamed accordingly. For example, the original implication is the converse of the converse; the contrapositive of the inverse is the converse, and so on. Definitions are implications for which the statement and its converse are both true. This is the real meaning of the reversibility of definitions. For this reason, definitions are sometimes called bi-conditional statements. Axioms and Theorems There are two kinds of if …, then… statements, axioms (also called assumptions or postulates) and theorems. Theorems can be proved to be true; axioms are assumed to be true without proof. A proof is a chain of reasoning starting from axioms, definitions, and/or previously proved theorems that convinces us that the theorem is true. (More on proof in a future post.) It would be great if everything could be proved, but how can you prove the first few theorems? Thus mathematical reasoning starts with (a few carefully chosen) axioms, and accepts them as true without proof. Everything else should be proved. If you can prove it, it should not be an axiom. Theorems abound. All of the important ideas, concepts, "laws" and formulas of calculus are theorems. You will probably see few, if any, axioms in a calculus book, since they came long before in the study of algebra and geometry. Learning Theorems When teaching students and helping them read and understand their textbook, it is important that they understand what a theorem is and how it works. They should understand what the hypothesis and conclusion are and how they relate to each other. They should understand how to check that the parts of the hypothesis are all true about the function or situation under consideration, before they can be sure the conclusion is true. For the AP teachers this kind of thing is tested on the exams. See 2005 AB-5/BC-5 part d, or 2007 AB-3 parts a and b (which literally almost no one got correct). These questions can be used as models for making up your own questions of other theorems. Like this: Definitions are similar to theorems, but are true in both directions; technically, this means that the statement and its converse are both true (). The double arrow is read "if, and only if." Both parts are either true or both parts are false. Definitions usually name some thing or some property. Definitions are not proved. The definition of continuity is a good example: A function f is continuous at x = a if, and only if, these three things are true (1) exist (i.e. is a finite number) (2) exist (i.e. is a finite number) (3) ("The limit equals the value.") "Play" with it: consider cases where only 2 of the 3 requirements are true – is the function still continuous? What would happen if you removed the requirements about finite numbers? To use a theorem one must be sure all the hypotheses are true. To use a definition one may say that either part is true once you have established that the other part is true. So if you know a function is continuous at a point, then the three statements are true; or if you can show the three statements are true, you may say the function is continuous. Here's an example: A typical AP problem might give a piecewise defined function and ask if it is continuous at the place where the domain is divided. To get credit for justifying an answer of "yes", students must show that all the requirements of the definition are met. Specifically, they must show that the limit as x approaches that point must equal the value of the function at that point (and both are finite). In turn, to show that this limit exist the student must show that the hypotheses of the theorem that says if the two one-sided limits are equal to the same number, then that number is the limit. To get credit for an answer of "no", the student must show that (only) one of the hypotheses is false. Finally, as with theorems, express definitions in words. With your students, "play" with the theorem or definition by making changes to the hypotheses and seeing how that affects the conclusion. Look at graphs. Don't just state the definition and expect students to understand it, remember it and use it correctly. Share this: Like this: Theorems are statements that summarize the results that are true in mathematics. Theorems are statements that have been proved true; but the emphasis in AP Calculus is not on proof. Rather, it is on what the theorems mean and how to use them. Theorems have two parts: the "if …" clause called the hypothesis and the "then …" clause called the conclusion. Students need to know both parts. In many theorems the conclusion is some sort of formula. The students need to know this, but also need to know when they can use it (the hypothesis tells them that). An early important theorem is the Intermediate Value Theorem (IVT). Take some time with this theorem. "Play" with it. The hypothesis requires that the function be continuous on a closed interval. Use graphs (sketches, no equation needed) to show cases where the conclusion is both true and false when the function is not continuous. Can the function take on values not between f(a) and f(b)? Can you find a case where the hypothesis is met, but the conclusion is false? (Let's hope not!) Consider the theorem (), its converse (), its inverse () and its contrapositive () by looking at graphs of each case. (For the IVT the converse and inverse are false. The contrapositive of any true theorem is also true.) Finally, for this and for all the important theorems that you use this year, express them in words, "play" with them by making change to the hypothesis, and look at graphs. Don't just state the theorem and expect students to understand it, remember it and use it correctly. The next post will be about definitions, which are similar to theorems in lots of ways. Share this: Like this: A blog for calculus teachers and students by Lin McMullin Thru the Year The listing of my posts from last August through November has been added to the "Thru The Year" page on the top navigation bar. I hope this will help you find the topics you want in each month a little ahead of when you teach them. This list will be updated monthly.
641769 / ISBN-13: 9780817641764 A Beginner's Guide to Graph Theory This beginner's textbook is intended for a first course in graph theory. It strikes a balance between a theoretical and practical approach, ...Show synopsisThis beginner's textbook is intended for a first course in graph theory. It strikes a balance between a theoretical and practical approach, consisting of carefully chosen topics to develop graph-theoretic reasoning for a mixed audience.Hide synopsis ...Show more welcomed by those new to the field

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.