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Free Kindle eBooks | Mathematics |
Showing Free eBook 1-22 of 22
1 review
by Jack Alexander McDonald
This book is in a series of two books both consisting of one hundred 11 plus maths questions. The first book The Foundation Questions has easier questions and the second book The Advanced Questions has the harder questions. These questions have been handpicked to help pupils pass their 11 plus Maths. These questions have been shown to enable children to succeed in passing the 11 plus maths exam. They have found them invaluable and I hope you will to.
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by Michael Andrew
"If an individual is not aware of a systematic, objective approach to their own efficiency systems, they are at a disadvantage to all those who are." Michael Andrew25 Years in the making, The Efficiency Playbook illustrates over 65 core foundations and tactics to readers allowing them to adapt and personalize into their own productivity game plan. The short, easy to digest chapters draw from the stories and experiences of the author Michael Andrew, as a Division-I footballer, missionary, scientist, photographer, independent film producer, disaster-aid responder, and serial entrepreneur, there is someone for everyone.The Efficiency Playbook includes examples and illustrations of:- Why multi-tasking is a myth and should be avoided- How to measure new career opportunities and which ones to ...
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by Tim Ander
Think Fast with Vedic Math Secrets and Mental Calculation Tricks!Read this book for FREE on Kindle Unlimited – Order Now!When you read Fast Math, you'll discover Vedic Math Techniques for mastering basic functions:AdditionSubtractionMultiplicationDivisionand so much more!With this fascinating guide, you can quickly and easily determine the square roots of perfect square numbers – and perform many other feats of mental gymnastics.These Vedic secrets mean you don't have to memorize mathematical facts anymore. By grasping the inner workings of math structures, you can make sense of all kinds of numbers – without a calculator or a computer!The written addition most of us learn in school relies on simple and slow systems like "carry the 1" to get answers. There is an alternative â...
by hridya bhatia
Become an absolute GENIUS!Calculate Faster than a calculatorMemorise names of clients teachers student or even all the Mughal Emperors!This book can be read by Anyone STUDENTS, TEACHERS, PARENTS, AND THE LIST GOES ON!...
1 review
by Wiley
The 2013 Statistics Reading Sampler includes select material from seven Wiley Statistics titles:An Accidental Statistician: The Life and Memories of George E. P. Box,Applied Logistic Regression, Third Edition,The Art of Data Analysis: How to Answer Almost Any Question Using Basic Statistics,Multiple Imputation and its Application,Data Mining and Business Analytics with R,High-Dimensional Covariance Estimation: With High-Dimensional Data,Introduction to Logistics Systems Management, Second Edition.For each selection, you'll find the full Table of Contents as well as the first chapter. To read more from these books, you can purchase the full book or e-book at your favorite online retailer.
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We find an expression for the multiple curl operator in matrix form. Author: dott. Alessio Mangoni, master's degree in physics....
by Iris Cooke
I am a mathematician who has studied the state lottos for numbers games and general studies of scratch-offs, and have written this book which contains numerous spreads for playing three-digit and four-digit games according to my own algorithms that I believe will produce more winners. The concept for these algorithms is explained in the book, as well as the way to choose scratch-off games that have the best odds of winning....
3 reviews
by John Slavio
Do you want to learn programming but too intimidated by the complexity?At some point of the time or the other, every computer guy starts to have a feeling of making a computer program. However, most of them never make a move towards this feel because computer programming sounds scary. In fact, computer programming isn't scary at all. All it takes is a correct selection of programming language to begin your journey as a computer programmer. Programming languages are made to make the human life better than before. These languages help in making programs which increase the overall productivity, communication, and efficiency of the work. Out of so many programming languages to choose from, python is one of the most loved programming languages among computer geeks. This is because python is o...
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by Steve Tale
Java: The Ultimate Beginners Guide to Java ProgrammingMore than anything, what you have to understand about Java is that it is a general purpose, object-oriented programming language- which means that it is easier to understand than other programming languages. It was designed primarily so that software developers could just code their programs once, and allow those programs run anywhere, or at least, in most platforms, as long as those platforms allow Java code to run in them. Java primarily uses some of the same elements that C+ and C++ use, but transforms them into much simple, easy to understand versions that could also be modified depending on the developer's preferences. Java has also been created to run with enough runtime support, both for hardware and software, by means of repre...
8 reviews
by Edwin Abbott
This special edition is a distinguished vintage reproduction, of the 1884 satirical novella Flatland, by the English schoolmaster Edwin Abbott. Meticulously elaborated by the editorial team of Chiron Academic Press in collaboration with the renowned literature publisher Edition l'Aleph ( this special edition, pays particular attention to the very authentic details of the editorial of text and images, fine type setting, mise-en-page, production, and print. The result is a revival of the vintage for the 21st century's reader. Thus a unique reading experience for the book lovers and collectors of this genre. A recommended edition to libraries.Writing pseudonymously as "A Square", the book used the fictional two-dimensional world of Flatland to comment on the hierarchy of Vict...
31 reviews
by Metin Bektas use the "Look Inside" feature. From the author of "Great Formulas Explained" and "Physics! In Quantities and Examples".
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2 reviews
by iCode Academy and Python Language
Are You Ready To Learn Python Easily? Learning f...
1 review
by Wiley
The 2014 Statistics Compilation Sampler includes selected material from six Wiley Statistics titles:Nonparametric Statistical Methods, 3rd EditionApplied Missing Data Analysis in the Health SciencesOnline Panel Research: A Data Quality PerspectiveApplied Linear Regression, 4th EditionWillful IgnoranceHow to Design, Analyse and Report Randomised Trials in Medicine and Health Related ResearchFor each selection, you'll find the full Table of Contents as well as the first chapter. To read more from these books, you can purchase the full book or e-book at your favourite online retailer. ...
We show the procedure and the calculus to find the solid angle of a cone with vertex known and of a sphere as a function of its distance. Author: dott. Alessio Mangoni, master's degree in physics....
7 reviews
by Leslie Copley
Introduction to and use of complex analysis and algebraic techniques to understand the solution of boundary value problems. Physics examples serve to introduce the fundamental partial differential equations and "special functions" of mathematical physics. A thorough analysis of Green's functions leads to a discussion of integral equations. Supplementary topics include dispersion relations and rational function approximation. ...
6 reviews
by Kapoo Stem
Daily Math Practice 75 Worksheets
This e-book contains several math worksheets for practice. There is one worksheet for each type of math problem including different digits with operations of addition, subtraction, multiplication and division. These varying level of mathematical ability activities help in improving adding, subtracting, multiplying and dividing operation skills of the student by frequent practicing of the worksheets provided.
There is nothing more effective than a pencil and paper for practicing some math skills. These math worksheets are ideal for teachers, parents, students, and home schoolers. The companion ebook allows you to take print outs of these worksheets instantly or you can save them for later use. The learner can significantly improve math kn...
This book presents the main ideas and techniques in the field of continuous smooth and nonsmooth optimization....
by Abjar Bahkou
The book presents Medieval Christian writer Gerasimus who in a debate with Islam presents reasoned proofs that refute charges of deception and duplicity made against Christians by means of logically constructed arguments about the being of God and His relationship to creation. He sets out the basic points of controversy and outlines a response in a form that would make an excellent introduction to Christian theology for the Muslim environment....
by Adrian King
It is a dark time for the Minecraft world in this final adventure. The epic war between human players and super mobs reaches a thundering crescendo: Steve aided by courageous volunteers, desperately battles to hold back the Mob invasion as the evil army of mobs tunnels into their dimension. In facing total annihilation, the Minecrafters of the last bastion of civilization fight not only for their lives, but for the future of the Overworld City itself | 677.169 | 1 |
Product Description
Vikatan Notes - Mathematics ( 10th Standard )
Dear Student,
We thank you for choosing this Vikatan Notes for Mathematics. The special features of this Notes are •One mark questions are taken only from the textbook. The question as well as the year in which they have appeared are given side by side. •Enough explanations have been given for the two mark questions with suitable formulae and pictures. •We have introduced ''Do it yourself" in each and every chapter for easy understanding and improving your problem solving skill. •Extra points and notes have been introduced to verify the solution. •Five mark questions have been presented with proper explanations, formulae and diagrams. •Ten mark questions have been designed to improve your drawing skills and get full marks for the questions. •Important and repeated questions have been marked with symbols. • The beginning and end of each solution have been underlined. We have prepared this Notes with great care and in a simple manner so that any student can easily understand the concept and score 100 100. | 677.169 | 1 |
Showing 1 to 30 of 48
MAT 123 Spring 2016
Name_
Kahn
Zhang
Recitation_
Wertz
_
Problem 1a (5 points):
1)
You have just been offered a job at Dash, Inc. Your starting pay will be
$38,000 a year on January 1, 2017, and each year you will receive a $4000 raise,
effective January
MAT 123 Introduction to Calculus
Final Exam
Wednesday 5/13 8:00am
Jason Starr
Spring 2015
MAT 123 Practice for Final Exam
Remark. The final exam will be cumulative. Please consult the review sheets for Midterms 1
and 2 in addition to this review sheet. If
MAT 123 - Graphing Practice
The book doesnt have many practice problems to help you graph rational, exponential, and
logarithm functions, so try to graph the following functions on your own. Include as much information in your graphs as you can - proper d
Syllabus for MAT 123 Spring 2016
MAT 123: Introduction to Calculus
About the Course
About this course: The goal of this course is to ensure that you have a proper background to
take calculus at Stony Brook. This means that we will need to accomplish sever
MAT 123 Spring 2016
Name_
Kahn
Zhang
Recitation_
Wertz
_
Problem 1 (5 points):
1)
You have been elected the CEO of Chump Co., a manufacturer of bad
hairpieces, and you need to restore the company to profitability. You run some
numbers and find that your r
UNIT TWO
Special Triangles
Now that we have learned the three basic trig ratios, lets learn how to find the sine,
cosine, and tangent of angles in special triangles.
You should remember from Geometry that an equilateral triangle has some
special propertie
UNIT THREE
Trig Ratios for Other Angles
Now lets learn how to find the trig values for other angles. Suppose that you
draw acircle of radius 1 (the unit circle), centered at the origin. Pick a point in
Quadrant I on the circle and draw the radius from the
UNIT ONE
The Basic Trig Ratios
Trigonometry consists of learning how to use six different functions, or ratios,
which show up in a surprisingly large number of places. Where do they come from?
A good place to start is with some basic geometry. Remember si
UNIT FIVE
The Reciprocal Functions
So far, we have been learning how to find the sine, cosine, and tangent of any
angle. Now it is time to learn three about more trig functions, which are called the
reciprocal functions. Each one of these functions is the
UNIT FOUR
Degrees and Radians
When we measure the size of an angle, we usually express that size in degrees. Now
we are going to learn another set of units to measure an angle. In the figure below,
we have a circle of radius 1 (the unit circle). The circu
Introduction to Calculus Advice
Showing 1 to 3 of 4
He's a great teacher who engages with the classroom. His teaching methods are fair and easily understandable
Course highlights:
I learned the process of calculus problems rather than just the questions and answers
Hours per week:
9-11 hours
Advice for students:
Manage your time wisely and remember when deadlines approach
Course Term:Fall 2016
Professor:david khan
Course Required?Yes
Course Tags:Great Intro to the SubjectGo to Office HoursGreat Discussions
Feb 25, 2017
| Would highly recommend.
Pretty easy, overall.
Course Overview:
Professor Flynn is willing to help students as much as she can and provides a thorough explanation of how to solve problems. It may seem like the class moves very quick but she provides a lot of examples that are really useful.
Course highlights:
MAT 123 is Pre-Calculus so most of what we learned in the beginning involved this such as slope, y=mx+b, and graphing. As the semester continues on we start brining in cos, tan, and sin and their derivatives.
Hours per week:
0-2 hours
Advice for students:
In order to succeed do the work. She assigns web assign and if you do the web assign, watch the videos, and practice you'll do well in the course. Writing down examples and going to lectures help as well.
Course Term:Fall 2016
Professor:MiriamFlynn
Course Required?Yes
Course Tags:Great Intro to the SubjectGo to Office HoursMany Small Assignments
Jan 16, 2017
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
The professor is by far the most fantastic professor I've had in a while. He cares so much about his students education and dedicates all of his time into his teaching so that he can watch his students succeed. He only wants the best for all.
Course highlights:
The highlights of this course was about introduction to calculus. We would go over sin, cosine, tangent, algebraic problems and calculus problems.
Hours per week:
9-11 hours
Advice for students:
Study a little every day, go to lecture, go to office hours and mostly don't be afraid to ask questions. Its tempting to skip because you want to sleep in. But in the end, when it counts, the amount of time and effort you put into this class will show and it will be worth every lecture you decided not to skip. | 677.169 | 1 |
Сollege math course syllabus (Reply)
This is the first course in the college-preparatory two-course sequence (MAT 0018 and MAT 0028) designed to prepare students for college-level mathematics courses. This course is a study of the basic skills and concepts of pre-algebra from the point of view of the college student who needs an understanding of pre-algebra. Major topics include operations with integers, fractions, decimals, percents, geometric figures and their measures (including application problems), and other pre-algebra topics.
This is the second course in the college-preparatory two-course sequence (MAT 0018 and MAT 0028) designed to prepare students for college-level mathematics courses. This course is a study of the basic skills and concepts of basic algebra from the view of a college student who needs an understanding of basic algebra. Major topics include operations on signed rational numbers, simple linear equations and inequalities in one variable, operations on polynomials (including beginning techniques of factoring), integer exponents, brief introduction to radicals, introduction to graphing, applications, and other basic algebra topics. | 677.169 | 1 |
math subjects are involved in game programming?
2 posts in this topic
I'm preparing to make a video course about maths. Not all mathematics in general but the specific concepts needed to program games. I want to help people get into programming and/or gamedev without the fear of maths. I'll try to do my best to explain things in a clear manner, show visual examples as much as i can etc.
The format of this course is my voice and the screencast of what I draw with my tablet. I may mix it with images and some very visualy stimulating math videos I can find (you can help here too).
But I want a solid syllabus that makes the course available to virtually all levels and doesn't leave any game or programming related math concept out.
This is what I have thought:
I will first teach "pre-algebra" concepts:
a relaxed overview of number theory (integers, floats..)
then order of operations
variables
equations and inequations
factors and prime numbers
fractions
percents
functions and graphs overview.
Then more algebra itself:
some equation solving
more complex function concepts (domain and range
graphs 2 sort of (linear functions, finding slope etc)
overview of system of equations and how to solve them
matrices
Not sure if talking about series (arithmetic, geometric..) or logarithms...
Introduction to function of circles and ellypses
Pre-calc:
more equations and graphs
more logarithms? (like their properties and such)
more functions (composition etc)
very basic geometry (radiant vs degrees etc) and Pitagoras
trigonometry
Each of those concepts will be a video-class of about 5-15minutes.
And I'm kind of stuck. Did I put too much in? Where should I introduce vectors? What else is needed for programming in general or gamedev in particular?
All ideas and suggestions are welcome
Thanks!
BTW: This will be a free course, doing it for the fun and to give back.
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But it is very difficult to make advanced games without advanced mathematics.
3D mathematics is linear algebra. If you don't have linear algebra you are going to have a nasty time doing much in a 3D world.
Then you can have more complex simulations. If your game physics is simple Newtonian math and you don't do anything fancy in response to collisions you can get away with a combination of trig and linear algebra. But anything more complex and you will want significantly more math, depending on the actions. Fluid simulations generally need math for fluid dynamics. Modern graphics require significant mathematics, be prepared to implement masters-level and PhD-level research papers; you don't necessarily need to understand the math to blindly implement the algorithms, but it absolutely helps.
Over the years I have heard many people say variations on, "I wish I had more math skills". I have only once heard someone complain about knowing too much mathematics, and that was because he was tired of people bothering him trying to have him break down the math into simpler tasks that others could understand.
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I would definitely include logarithms, if only to grant more understanding to algorithms with O(log n) and O(n log n) complexity, since those are an incredibly important part of programming theory.
Variables is always a crazy concept when trying to teach algebra within the context of programming. That was a huge breakthrough moment for me when i realized that "x = 5" is radically different for algebra than it is for programming. Equations in algebra versus conditionals in programming can be a similar source of confusion.
For fractions and percents, delving into the subject of rounding methods can be important for programmers, whereas rounding tends to be treated as an advanced tangential subject within math. Mostly because standard mathematical techniques tend to have difficulties dealing with rounding, whereas within a programming context, rounding is no big deal.
I consider mathematical functions to be most relevant to programming when thinking about how to map some combination of input values to an output value, and achieve a certain type of behavior. This shows up a lot in game mechanics where miscellaneous game variables influence the value of another game variable, and also in animation, where time and certain attributes of an animation affect the visible properties of the animation.
Not sure about factors and prime numbers. I guess they're relevant for cryptography, but that's not really a big deal for most game programmers. Maybe in terms of writing quick 'n' dirty hash functions and pseudo random number generators, though.
For trigonometry, it's good that you cover it after linear algebra. It would probably be useful to make sure you emphasize that most of the time, if a game programmer is using trigonometry directly in code, there's a very good chance they'd be better off using linear algebra; they just gotta think about the problem from a different angle. (Heh, a pun.) | 677.169 | 1 |
Solid Geometry, by Harry Kretz
"This little volume presents a rich and ready compendium for grade eight on Platonic solids, together with a practical and lively teaching approach, which will be welcomed by novice and old hand alike. And the teacher will find that the development set forth in this manual will enable him or her to effect the task at hand in a manner which is pedagogically effective and mathematically meaty, lucid, and meaningful."
- Amos Franciscelli, from the Foreword
The first part of this manual is a collection of plates as they might appear in a student's notebook. The second part is a teacher's guide describing some of the class activities that lead to the notebook.
Harry Kretz taught high school math for many years at Hawthorn Valley Waldorf School in Ghent, New York. Since retiring from full-time work, he has worked more than full-time travelling to teach courses for a few weeks at a time in various Waldorf schools and teacher training centers. | 677.169 | 1 |
Finite Mathematics – Lial, Greenwell, Ritchey – 8th Edition
Widely known for incorporating interesting, relevant, and realistic applications, this text offers many real applications citing current data sources. There are a wide variety of opportunities for use of technology, allowing for increased visualization and a better understanding of difficult concepts. MyMathLab, a complete online course, will be available with this text. For the first time, a comprehensive series of lectures on video will be available.
Features:
Section Opening Questions are thought-provoking questions which are revisited again and answered within the section.
Variety of Applications—using real data from current sources, applied exercises are grouped by subject and highlighted for easy identification; Extended Applications appear at the end of most chapters.
Excel Spreadsheets are included in examples and exercises as appropriate, allowing students to work problems that closely relate to real-life and business situations.
Optional Graphing Calculator Integration. Graphing calculator discussions and TI-83/84 Plus screens are used in many examples.
Exercise Sets. Includes exercises from Japan's university entrance exam showing how math is used in a variety of areas and around the globe. In addition, there are exercises that require writing and conceptualization in order to promote critical thinking, deeper understanding, and to integrate concepts and skills. Connection Exercises, denoted with an icon, integrate topics/concepts from different sections.
Chapter R, Algebra Reference begins the text allowing students to brush up on their algebra skills | 677.169 | 1 |
9780805811Integrating Research on the Graphical Representation of Functions (Studies in Mathematical Thinking and Learning Series)
This volume focuses on the important mathematical idea of functions that, with the technology of computers and calculators, can be dynamically represented in ways that have not been possible previously. The book's editors contend that as result of recent technological developments combined with the integrated knowledge available from research on teaching, instruction, students' thinking, and assessment, curriculum developers, researchers, and teacher educators are faced with an unprecedented opportunity for making dramatic changes.
The book presents content considerations that occur when the mathematics of graphs and functions relate to curriculum. It also examines content in a carefully considered integration of research that conveys where the field stands and where it might go. Drawing heavily on their own work, the chapter authors reconceptualize research in their specific areas so that this knowledge is integrated with the others' strands. This model for synthesizing research can serve as a paradigm for how research in mathematics education can -- and probably should -- proceed | 677.169 | 1 |
Making Sense of Mathematics for Teaching High School: Understanding How to Use Functions
Description: Develop a deep understanding of mathematics by grasping the context and purpose behind various strategies. This user-friendly resource presents high school teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. Explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success. Combine student understanding of functions and algebraic concepts so that they can better decipher the world. Benefits Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics. Explore how to develop, select, or modify mathematics tasks in order to balance cognitive demand and engage students. Discover the three important norms to uphold in all mathematics classrooms. Learn to apply the tasks, questioning, and evidence (TQE) process to ensure mathematics instruction is focused, coherent, and rigorous. Gain clarity about the most productive progression of mathematical teaching and learning for high school. Watch short videos that show what classrooms that are developing mathematical understanding should look like. Contents Introduction Equations and Functions Structure of Equations Geometry Types of Functions Function Modeling Statistics and Probability Epilogue: Next Steps Appendix: Weight Loss Study Data References Index" | 677.169 | 1 |
The meeting will focus on actual or possible applications of nontrivial
computer algebra techniques to other fields and substantial interactions
of computer algebra with other fields.
For detailed information on the conference we refer to the Websites | 677.169 | 1 |
Pages
20 May 2017
Calculators for the New A Level
In September over 100 students at my school will be starting the new A level course. I've been trying to find out exactly what calculator they will need and how they can get the best deal.
I'll say upfront that I am most definitely not a calculator person. Some maths teachers get really excited about calculators. I don't. I lost my lovely 20-year-old calculator last year so bought the Casio 991EX ClassWiz at #mathsconf8 in October. I've only used it for standard calculations so far, and my main thoughts are: (a) the font is weird (b) the menus are quite user-friendly and (c) the white case gets dirty quickly. That's about it. People who love calculators seem to really love the ClassWiz. It has some neat features - if you're interested, this review on Amazon gives some insight into the functionality that people are getting excited about.
The ClassWiz is not the only calculator that's suitable for the new A level (do check out the TI-30X Pro too). But I have a feeling that the ClassWiz will be the one that most new A level students are told buy in September, which is why I'm focusing on the ClassWiz in this post.
Around 100,000 students will each spend over £20 on a new calculator this September. £2,000,000 spent on calculators is a really big deal. So before our students collectively give Casio this vast amount of money, I need to be sure that it's absolutely necessary.
"Ofqual's subject-level conditions and requirements for Mathematics and Further Mathematics state that calculators used must include the following features:
an iterative function
the ability to compute summary statistics and access probabilities from standard statistical distributions
the ability to perform calculations with matrices up to at least order 3 x 3 (FM only)
For the 2017 A levels students will require a calculator that can calculate Binomial and Normal probabilities directly from values. The minimum standard for this is an advanced scientific calculator, such as the Casio 991EX ClassWiz or the TI-30X Pro..."
Just to clarify - A level maths students will probably already have a calculator from GCSE that does everything they need - except binomial probabilities. This is the one thing that they will need to buy a new calculator for.
Everything else that the ClassWiz does that current calculators don't do is a 'nice to have' for the new A level but not essential. It does do some cool stuff, but bear in mind that the extent to which 'nice to have' functionality is used depends heavily on whether teachers know how to use the functionality themselves and have enough time to teach it to their students. My understanding is that timing for the new A level is going to be really tight as it is (my school has nine hours a fortnight at A level and I'm told that it probably won't be enough time to get through the content). Given time constraints and huge class sizes, I can't see that I'll be spending much (if any) time on any non-essential calculator skills.
We're told that there is now a 'requirement for the use of technology to permeate teaching and learning' at A level. I'm a big fan of using Desmos in lessons - most A level maths teachers have been doing this for years anyway. Desmos is free, easy to use and works well on students' phones. The large data set work will probably be done in Excel or Geogebra so I guess I'll be booking IT rooms for that when the time comes (which is easier said than done!).
People who are looking to make money from calculator sales might try to convince teachers that graphical calculators are a requirement for the new A level. This is misleading. I'll stick with Desmos. Graphical calculators do offer some benefits to students but even the newest models are dated and unintuitive. The article "Pricey Graphing Calculators Could Be Headed for Extinction" is worth a read. In many schools this expensive equipment ends up sitting unused in a cupboard after a year or two. However, if you're skilled at using graphical calculators and you have the time to teach your students how to use them properly, then that's great - by all means buy them for your students (they're expensive so this unlikely to be an option in large schools) or ask students to buy one themselves (probably only an option in private schools).
So, in summary, for A level maths it is essential that students buy a new calculator, purely for binomial probabilities, and the ClassWiz is a sensible choice for most students.
Where to buy
On A level induction day next month, I'll tell my students that they will have to buy a new calculator in September (once they've confirmed they are definitely taking maths). I would like them to buy their calculators through high street retailers.
Presumably the first month or so of maths A level will focus on non-calculator topics that were previously in C1, so October half-term might be a reasonable deadline for students to buy their new calculator.
Casio tells me that the ClassWiz will hit retailers in 'maybe September', but probably at a higher price than they are currently on sale for. A bit more certainty on dates would be helpful - this is all a bit too last minute for me. I'm frustrated by Casio's approach here. I very much hope that Casio has enough stock to cater for huge levels of demand in September.
Casio warns to avoid buying the ClassWiz from Amazon at the moment because 'they sell non UK imports'. The ClassWiz currently being sold on Amazon for £32.50 comes with foreign language instruction manuals.
Many schools are buying calculators in bulk for their teachers to use, or to sell to their students. Sources of calculators include:
There may be discounts for bulk orders. VAT can be reclaimed if the calculators are for school use, but not if sold to students.
I'm reluctant to buy in bulk and sell to students because we've had nightmares with this in the past (does anyone want to buy three unopened boxes of brand new C2 textbooks from us? Didn't think so). I'd rather students took responsibility for their own calculator purchase.
Support
Dr Frost is an absolute superstar and has created a brilliant free tool for training staff and students in how to use the ClassWiz. It's a PowerPoint guide explaining every key and mode.
Casio offers an emulator, but the licence is £9.95 + VAT per year per computer.
If you want your team to be trained on how to use all the functionality on the ClassWiz, perhaps speak to your local Maths Hub. This certainly seems like something the Maths Hubs could usefully offer in July and September if they have the expertise. The FMSP is offering numerous free calculator events but these sessions focus on graphical calculators, not the ClassWiz.
Profiting...? Calculator suppliers are not the only companies cashing in on the change to A levels and GCSEs. Textbook publishers are benefiting too. With such limited funds in education - redundancies, growing class sizes and leaking roofs - this is a frustrating use of public money. Curriculum change is an expensive business.
People have said to me that 'kids these days' don't think twice about buying the latest iPhone so £30 for a calculator isn't a big deal. Perhaps it's not a big deal on an individual basis, but I'm looking at the bigger picture - over £2 million. That's a big deal.
11 comments:
I agree with you on many points. I don't use calculators often. Mathematics is great because you only need paper and pen. When I have to do serious calculations or make graphs, I'll use a computer and dedicated software, such as desmos, matlab, R, mathematica, ... So, I don't understand this new requirement. I checked the price of the TI-30X-Pro and it seems much cheaper than the Casio (£16.45 on amazon). Why would you choose the Casio then?
Good question! The people I know who love calculators talk a lot about the ClassWiz, but I don't know if they've done a comparison. It would be interesting to find out why it's considered the better option. Perhaps it's just because in the UK Casio massively dominates the market - most GCSE students currently own a Casio.
Some calculator companies offer a sale or return option so you can buy the calculators with a bulk discount, sell to students and then return the rest. At least this will mean you don't end up with loads left over
We currently get them to buy a graphical calculator. We get them to order through us making sure we get the money first then do a massive order. Have always managed to get the cheaper rate that way and we are never out of pocket.
Having tutored a private school student who was great at using a graphical calculator but had no understanding of the underlying maths, I'm still not convinced about them. I know that they work well for some teachers in some schools, but they definitely don't suit everyone.
I recently attended an event where I got to play with the latest graphical calculator and I thought it was awful!
I taught IB for five years and Graphical Display Calculators were compulsory and necessary. Therefore, much more time could be spent on choosing the correct procedure and interpreting the result. Loved them and embedded them in all lessons. The materials from IB are out there and easy to use. I think they are well worth a look.
We applied for funding and had a class set which was used from Year 8. Nearly all sixth form maths students then chose to buy their own as they realised how useful they are. £80 is pricey, but was often an Xmas present from grandparents! | 677.169 | 1 |
Topology: A First Course
Hardcover | December 28, 1999
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This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. Includes many examples and figures. GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness and Compactness. Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness. Complete Metric Spaces and Function Spaces. Baire Spaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The Fundamental Group. Separation Theorems. The Seifert-van Kampen Theorem. Classification of Surfaces. Classification of Covering Spaces. Applications to Group Theory. For anyone needing a basic, thorough, introduction to general and algebraic topology and its applications. | 677.169 | 1 |
Topics:
Introduction
Sequence and series is a mathematical concept that draws majorly from the basic number system and the simple concepts of arithmetic. This is the reason that makes it an important topic for this exam. On an average, 1–3 questions have been asked from this topic in the CAT almost every year in the last 12 years. Besides the CAT, this topic is important for other exams like IIFT, SNAP, XAT, MAT. One good thing about the problems from this chapter is that they can be solved simply by the application of logic or some very simple concepts of calculation | 677.169 | 1 |
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2 MB|24 pages
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Included in this package is a complete set of guided notes and answer key for an algebra unit dealing functions. This unit provides students with an overview of the basics of functions which are built upon all throughout the algebra course as students study individual families of functions. Lessons include domain/range (interval vs. inequality notation), function/relation, function notation, operations with functions, compositions of functions, introduction to parent functions (constant, linear, quadratic, cubic, absolute value, and radical), and transformations of functions. Included in each lesson are "You Try!" questions to be used for formative assessment. I typically go around the room and put a stamp or sticker on each student's paper when they correctly answer the question. At the end of the packet is learning target checklist where students can rate themselves each day on their progress toward mastery of the objectives Some equations require MathType 6 to edit. Individual lesson notes and related products are coming soon. | 677.169 | 1 |
£24 text in linear algebra which is intended for a one-term course. It examines the relation between the geometry and the algebra underlying the subject. It features sections on linear equations, matrices and Gaussian elimination, vector spaces, linear maps, scalar products, determinants, and eigenvalues. | 677.169 | 1 |
Created through a "student-tested, faculty-approved" review process, MATH APPS is an engaging and accessible solution to accommodate the diverse lifestyles of today's learners at a value-based price. The book's concept-based approach, multiple presentation methods, and interesting and relevant applications keep students who typically take the course--business, economics, life sciences, and social sciences majors--engaged in the material. An innovative combination of content delivery both in print and online provides a core text and a wealth of comprehensive multimedia teaching and learning assets, including end-of-chapter review cards, downloadable flashcards and practice problems, online video tutorials, solutions to exercises aimed at supplementing learning outside of the classroom. | 677.169 | 1 |
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Description
Description
Excerpt from Graded School Arithmetic, Vol. 2: An Elementary Text for Use in Public and Private Schools, From the Fifth to the Eight Year, Inclusive Text-books on arithmetic have been criticised by teachers and school supervisors for several reasons, among which are the following - 1. They contain unnecessary matter, by giving undue emphasis to impractical topics. 2. The gradation and pedagogical presentation of the subject is often made secondary to the logical arrangement of the material. 3. The many subdivisions of arithmetic, arising from the multitude of applications, often lead to a confused conception of the subject as a whole. Emphasis should be given rather to the few principles of relation, than to an extensive outline of subjects. 4. There is often too much discussion, and too little work. 5. Arithmetic is often considered as something apart from algebra and geometry, and effort is made in the grammar grades to show its proper relation to those subjects. 6. The subject has too little relation to business life. The Graded School Arithmetic, Book Two, has been prepared to meet the above-named criticisms. 1. But little emphasis has been given to compound and annual interest, partial payments, foreign exchange, solid mensuration, and other topics, and thus much unnecessary matter has been eliminated. 2. The matter has been presented with a view to the gradual development of the pupil's mental power. 3. The several topics are related by the emphasis given to common principles. For example, Ratio and Simple Proportion are introduced before Percentage and Interest, because the principles of Ratio and Proportion are fundamental to the operations of Percentage and Interest | 677.169 | 1 |
-- This new MyMathLab-based course option from Lial/Hornsby/McGinnis offers a complete intermediate algebra course with embedded review of prerequisite topics from previous courses. The Integrated Review MyMathLab course model can be used to bring underprepared students up to speed, helping to address the challenge of varying skill levels with one seamless MyMathLab course.
Integrated Review MyMathLab courses provide the full suite of supporting resources for the main course content, plus additional assignments and study aids for students who will benefit from remediation. Assignments for the integrated review content are preassigned in MyMathLab, making it easier than ever to create your course! About Lial/Hornsby/McGinnis's Beginning Algebra)-new study skills activities in the text, an updated and expanded Lial Video Library (available in MyMathLab), and a new accompanying Lial Video Library Workbook (available in MyMathLab).
0134196163 / 9780134196169 Beginning Algebra Plus NEW Integrated Review MyMathLab and Worksheets--Access Card Package Package consists of: 0134197089 / 9780134197081 Worksheets for Integrated Review for Beginning Algebra with Integrated Review 0321431308 / 9780321431301 MyMathLab -- Glue-in Access Card 0321654064 / 9780321654069 MyMathLab Inside Star Sticker 0321969332 / 9780321969330 Beginning Algebra | 677.169 | 1 |
AFM
Courses
A course on using basic arithmetic operations to solve problems in a variety of contexts. Topics include: whole numbers, fractions, percents, proportional reasoning, and simple linear equations. Students must earn a minimum grade of C to pass this course.
Prerequisites: Appropriate placement test score
View Sections
AFM 091 Pre-Algebra 1 2 Credits. 2 Lecture Hours. 1 Lab Hour.
A course on using basic arithmetic operations to solve problems in a variety of contexts. Topics include: integers, fractions, decimals, percents, variables, and algebraic expressions. Students must earn a grade of C or higher to pass. This course is delivered in a 7-week schedule.
Prerequisites: Appropriate placement test score
View Sections
AFM 092 Pre-Algebra 2 2 Credits. 2 Lecture Hours. 1 Lab Hour.
A course on solving problems with algebra. Topics include: algebraic expressions; linear equations of one and two variables; graphing; slope and rate of change; and algebraic, graphic, and numerical representation. Students need a graphing calculator. Students must earn a grade of C or higher to pass. This course is delivered in a 7-week schedule.
Prerequisites: AFM 091 (minimum grade C) or AFM 090 (minimum grade C) appropriate placement test score
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A course on mathematical modeling and problem solving. Topics include: polynomial, exponential, radical, and rational functions; and inequalities. Students need a graphing calculator. Students must earn a grade of C or higher to pass. This course is delivered in a 7-week schedule.
Prerequisites: AFM 094 (minimum grade C), or appropriate placement test score
View Sections
AFM 098 First Year Special Topics in Academic Foundations: Math 1-9 Credits.
A course on selected topics related to Academic Foundations: Math, which gives students opportunities to study information not currently covered in other courses. Grades issued are A, B, C, D, or F.
Prerequisites: Vary by section
View Sections
A project related to Academic Foundations: Math that is completed by one or more students to meet specific educational goals. Projects must have prior approval and supervision by AFM faculty. Grades issued are Satisfactory or Unsatisfactory.
Prerequisites: Vary by section
View Sections | 677.169 | 1 |
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Engineers and computer scientists who need a basic understanding of algebra will benefit from this accessible book. The sixth edition includes many carefully worked examples and proofs to guide them through abstract algebra successfully. It introduces the most important kinds of algebraic structures, and helps them improve their ability to
...Matrix algebra is one of the most important areas of mathematics for data analysis and for statistical theory. This much-needed work presents the relevant aspects of the theory of matrix algebra for applications in statistics. It moves on to consider the various types of matrices encountered in statistics, such as projection matrices and... | 677.169 | 1 |
The year-long program Matter and Motion included the study of calculus and physics throughout the year, chemistry in fall and winter, and a quarter of introductory analog circuits in spring. This all-level program covered introductory topics in these subjects through lectures, workshops, and labs. Students with previous college level course work in chemistry could opt of that part of the program. Students improved their mathematical and scientific reasoning and their problem solving abilities in calculus, chemistry, and physics. Student evaluations were based on quizzes, exams, homework, lab notebooks, a portfolio of collected work, and engagement in lectures, problem-solving workshops, and laboratories.
Program objectives for students included: improving ability to articulate and assume responsibility for their own work; improving oral and written communication skills; learning single- and multi-variable differential and integral calculus and their applications, particularly to physics; utilizing mathematical models that describe and explain motion in the natural world; using the main ideas of classical mechanics, waves, basic electrostatics, and parts of modern physics to solve fundamental and applied problems; learning foundational concepts in chemistry and using fundamental laboratory tools to investigate chemical and physical properties of solids, liquids, and gases; and an introduction to the construction and use of analog circuits, including op-amp circuits.
Calculus I with Laboratory: First-quarter (differential) calculus was covered, using Hughes-Hallet's Calculus: Single and Multivariable (6th ed). Students worked through chapters 1-4 in that textbook, including the concepts and definitions of limit and derivative, particularly in connection with motion; differentiation techniques including derivatives of trigonometric functions and implicit differentiation; and applications of differentiation including optimization and related rates. Students participated in six computer labs to support their conceptual understanding through visualization through considerable use of the online Desmos tool. Students submitted homework via the online system WileyPLUS and took quizzes as well as a midterm and final exam.
Calculus II: Winter-quarter (integral) calculus covered parts of chapters 5-8 from the same text, including the definite integral, construction of antiderivatives, methods of integration (including substitution and integration by parts), and applications of the definite integral to calculating work, centers of mass, volumes, and moments of inertia. Taylor series were also covered briefly. Student achievement was assessed via weekly problem sets of 10-15 problems, as well as midterm and final exams.
Calculus III: Spring-quarter calculus focused on multivariable calculus, covering chapters 12 – 16 (from the same text) including: functions of multiple variables; vectors; differentiating and integrating functions of several variables and applications; and transforming between coordinates systems, with particular emphasis on applications in the physical sciences. Student submitted online drill sets via WileyPLUS as well as written problem sets, and took quizzes, a midterm, and final exam.
University Physics I with Laboratory: Fall quarter focused on classical mechanics. Students learned to understand concepts about and solve problems involving: kinematics (translational and rotational), dynamics (force and torque), and conservation principles (momentum, energy, and angular momentum). These were reinforced by lab exercises frequently using Vernier LoggerPro software and sensors. Students worked through chapters 1-12 in Knight's Physics for Scientists and Engineers with Modern Physics (3rd ed.) and submitted homework via the online system MasteringPhysics. Students took quizzes, as well as a midterm and final exam.
University Physics II with Laboratory: Winter quarter coverage involved primarily oscillations, waves, and introductory electrostatics. Coverage (also in the text by Knight) included chapters 13 (Newtonian gravity), 14 (oscillations), 20-22 (waves), and parts of 25-30 (focusing on the electric field and electric potential, primarily of point charges and in the parallel plate capacitor), as well as Ohm's law and resistance. Labs included diffraction (single and multiple-slit), introductory DC circuits, and related elements from chemistry (galvanic cells, thermochemistry). Students completed weekly problem sets of 12-15 problems, as well as midterm and final exams.
University Physics III with Laboratory: In spring quarter, the program covered chapters 31-35, and 37-38 of the Knight text, including the fundamentals of circuits, the magnetic field and electromagnetic induction (including Faraday's law), AC circuits, and a brief overview of electromagnetic waves. The final two weeks included an introduction to modern physics, particularly quantization, the hydrogen atom, and the particle in a box. Students completed weekly problem sets of 10-14 problems, as well as midterm and final exams.
Introductory Analog Circuits Lab: Spring quarter included a weekly 2-hour lecture and 4-hour analog circuits lab. Texts were Horowitz and Hill's The Art of Electronics, 2nd ed., Chapters 1 and 4 (for ch. 4, we covered §4.01-4.09 on basic op-amps circuits, and parts of §4.11 on op-amp limitations) and labs from the affiliated lab manual by Horowitz and Hayes. In addition to the lab, there were weekly problem sets of 5+ problems, as well as an in-class, closed book midterm and final
Your graded exams (and note sheets) are available to pick up outside my office, Lab 2 3255. They are in alphabetical order, so please preserve that. Unless you have specific permission from that person, please do not pick up an exam on someone's behalf.
There were 100 points available on the exam. However, calculate your percent out of 90 points. If you earned 54 points or less (i.e. 60% or less when calculated out of 90 points), you are particularly invited to submit an Exam Revision, due to my office by 5 pm Mon. June 6; any one may submit an exam revision.
]]>Visiting physics faculty candidates on campus Fri. June 3 and Wed. June 15
Sat, 28 May 2016 22:23:35 +0000 reading →]]>We have invited 2 candidates to campus to interview for the visiting physics faculty position in next year's Matter & Motion. We hope that you might be able to attend some part of their campus visits. Each candidate will give short teaching demonstrations on topics provided to them in advance, and will also be available over lunch.
The first candidate will be on campus Friday June 3. The teaching demonstration will be from noon – 1, Sem 2 C2107. The candidate will then be available from 1 – 2 in the faculty/staff lounge in the basement of the CAB.
The second candidate will be on campus Wednesday June 15. The teaching demonstration will be from noon – 1, Sem 2 A2109. The candidate will then be available from 1 – 2 in the faculty/staff lounge in the basement of the CAB.
As you have just completed Matter & Motion, you are in a particularly good position to give feedback on the physicist to teach in that program next year. Please attend if you can.
On a related note, we will running hiring processes for permanent faculty in mathematics and in physics next year, to start at Evergreen in fall 2017. Please let Krishna know if you would be interested in serving on those search committees.
In case you misplaced it or missed getting it in class, you can find a blank (updated) version of the Calculus III Self-Check Worksheet here.
It is due Wednesday June 1 by 5 pm to Krishna's office, Lab 2 3255; leave them in the homework box.
The graded Exam 2 will be available to pick up from Krishna's office by 9 am Tuesday May 31 Wednesday June 1.
]]>All class meetings on Friday May 27 are in Sem 2 E3105
Thu, 26 May 2016 04:59:16 +0000 M&Ms! On Friday May 27, all your class meetings are in Sem 2 E3105. So you'll meet there in the morning for physics lecture, and also at 1 pm for the calculus final exam.
]]>Calculus III Final Exam, Fri. May 27 at 1pm in Sem 2 E3105
Thu, 19 May 2016 07:27:54 +0000 reading →]]>Due to the Memorial Day Mon. May 30 holiday, the week 30 schedule is constrained. To avoid having three exams in one week where one exam would have to be on the final day of our year together, we will have the cumulative Calculus III final exam on Week 9 Fri. May 27 at 1pm in Sem 2 E3105.
May use a personally prepared notesheet on an 8.5 inch x 11 inch piece of paper (both sides ok), a calculating device, and writing tools.
The Application also asks for information from the final summative evaluation, but notes that students currently in Matter and Motion may use interim evaluations. Again, several students have made missteps here. If you don't have your interim evaluation from fall and/or winter, please email Krishna directly chowdark@evergreen.edu and he can provide it to you. | 677.169 | 1 |
Math 321 is designed for students of
science and engineering to acquaint them with the potentialities of
the modern computer for solving the numerical problems that will
arise in their professions. It will give the students an opportunity
to improve their skills in programming and in problem solving.
Week No.
Date
Subject
1 & 2
Feb. 13-Feb 22
1.1 Review of Calculus
1.2 Binary Numbers
1.3 Error Analysis
3 & 4
Feb. 25-March 8
2.1 Iteration for Solving x=g(x)
2.2 Bracketing Methods for Locating a
Root
2.3 Initial Approximation and
Convergence Criteria
2.4 Newton-Raphson and Secant Methods
5 & 6
March 11-March
22
First exam
20 March
3.1 Introduction to Vectors and
Matrices
3.2 Properties of Vectors and Matices
3.3 Upper-triangular Liner Systems
3.4 Gaussian Elimination and Pivoting
3.5 Triangular Fractorization
7 & 8
March 25-April 5
4.1 Taylor Series and Calculation of
Functions
4.2 Introduction to Interpolation
4.3 Lagrange Approximation
4.4 Newton Polynomials
April 1-2
Midterm Exam
9
April 8- April
12
5.1 Least-squares Line
5.2 Curve Fitting
10
April 15- April
19
6.1 Approximating The Derivative
6.2 Numerical Differentiation
Formulas
11
April 22- April
26
7.1 Introduction to Quadrature
7.2 Composite Trapezoidal and
Simpson's Rule
12 & 13
April 29- May 10
Second exam
20 April 29
9.1 Introduction to Differential
Equations
9.2 Euler's Method
9.3 Heun's Method
9.4Taylor Series Method
9.5 Runge-Kutta Methods
14
May 13- May 17
10.1 Hyperbolic Equations
10.2 Parabolic Equations
10.3 Elliptic Equations
15
May 20- May 27
11.1 Homogeneous Systems: The
Eigenvalue Problem
11.2 Power Method
Homework and computer
assignments will be distributed for each section. | 677.169 | 1 |
Explore the main algebraic structures and number systems that play a central role across the field of mathematics
Al—linear algebra, abstract algebra, and number theory—into one comprehensive and fluid presentation, facilitating a deeper understanding of the topic and improving readers' retention of the main concepts.
The
Interesting applications are provided throughout to demonstrate the relevance of the discussed concepts. In addition, chapter exercises allow readers to test their comprehension of the presented material.
Al
"Sinopsis" puede pertenecer a otra edición de este libro.
About the Author:
MARTYN R. DIXON, PhD, is Professor in the Department of Mathematics at the University of Alabama, Tuscaloosa. He has authored more than sixty published journal articles on infinite group theory, formation theory and Fitting classes, wreath products, and automorphism groups.
LEONID A. KURDACHENKO, PhD, is Distinguished Professor and Chair of the Department of Algebra at the Dnepropetrovsk National University (Ukraine). Dr. Kurdachenko has authored more than 150 journal articles on the topics of infinite-dimensional linear groups, infinite groups, and module theory.
IGOR YA. SUBBOTIN, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University (California). Dr. Subbotin is the author of more than 100 published journal articles on group theory, cybernetics, and mathematics education.
Review:
"The book is well-written and covers, with plenty of exercises, the material needed in the three aforementioned courses, albeit in a new order." (Zentralblatt MATH, 1 December 2012)
"However, instructors contemplating such a unified approach should give this book serious consideration. Recommended. Upper-division undergraduates through researchers/faulty." (Choice , 1 April 2011). Hardback. Estado de conservación: new. BRAND NEW, Algebra and Number Theory: An Integrated Approach, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin,-linear algebra, abstract algebra, and number theory-into one comprehensive and fluid presentation, facilitating a deeper understanding of the topic and improving readers'9780470496367
Descripción John Wiley and Sons Ltd, United States, 2010. Hardback. Estado de conservación: New. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible.ZV9780470496367 | 677.169 | 1 |
IB Mathematics SL
IB Mathematics SL
Getting used to the rigor of the material and IB-style questions can be challenging, especially for a fast-paced course like Mathematics SL. Tokyo Academics is offering a specialized International Baccalaureate course in Mathematics Standard Level.
Guidance on IB Internal Assessments: All IB Math SL students hand in a math project for assessment to the IB, worth 20% of the final IB Mathematic course grade. Many schools require students to hand in 2 or 3 of these as practice.
Schedule
Class meets twice a week on Sunday mornings and Wednesdasy evenings. Sessions begin January 15th and continue until the end of the Semester in June 2017. | 677.169 | 1 |
DESCRIPTION: Math II continues a progression of the standards established in Math I, including expressions in the real number system, creating and reasoning with equations and inequalities, interpreting and building simple functions, expressing geometric properties and interpreting categorical and quantitative data. In addition to these standards, Math II includes: polynomials, congruence and similarity of figures, trigonometry with triangles, modeling with geometry, probability, making inferences and justifying conclusions.
TEXTBOOK: Pearson Geometry, 2014. $100.00
Course
Outline:
Units
Teaching Time
I.Transformations
10 days
II. Quadratic Functions
20days
III.Square Root and Inverse Variation Functions
15 days
IV. Similarity and Congruence
12 days
V. Trigonometry (Solving Right Triangles)
13 days
VI. Probability
10 days
STUDENT GRADE: The final average for the each 9 weeks will be computed using the following formula:
Homework 25%
Quizzes 25%
Major Test Average 50%
The final average for the semester will be computed using the following formula:
Blue or Black Pens—no red/pink/ purple ink on assignments will be accepted
Loose Leaf Paper—or notebook paper with the rings torn off
Folders with pockets to keep returned papers organized.
************NEW BOARD POLICY IN EFFECT************
Any student missing 6 or more days from a class will NOT receive credit for the class. This is true regardless of the nature of the absence (unexcused or excused).
CLASS RULES:
Come to class prepared.
Work quietly and remain in your seat, unless otherwise directed.
No sleeping, eating, or drinking in class.
Be in your seat and working when the bell rings.
Respect others and their property.
All Southern Wayne High School and Wayne County School System policies/rules are also in effect for this class
POLICIES AND PROCEDURES:
Students may be asked to work together for certain assignments. At these times, students should talk quietly, work together, share materials and ideas, and stay on task.
The majority of the student's work will be done independently. No form of communication or use of unauthorized resources during quizzes or tests will be tolerated.
Students are encouraged to ask questions and participate in discussions during class. Students should not interrupt, argue, or ignore any other person.
All work on homework, quizzes, or tests must be shown to receive credit. Answers only are not acceptable.
It is the student's responsibility to find out what assignments were missing when absent and to get the missed notes.
Absences will not excuse a student from turning in previously assigned work on the day he/she returns to class.
Any student that has exceeded 6 unexcused absences and/or is failing a class (59 or below) will not be allowed to miss that class by leaving school during regular instruction for any school-sponsored function, academic (field trip) or athletics.
Students who have outstanding make-up work will receive a grade of "2" as a place holder until the work is made up. Once work is made up and submitted, the "2" will be replaced by the proper grade. Work not made up in 5 days will become a "0".
DRESS CODE POLICIES
Shorts, skirts, dresses, or other clothing cannot be more than 3 inches above the top of the knee, including when leggings or tights are worn.
Shirts and tops should be long enough to cover the midriff when sitting or standing.
Shirts, tops, and dresses must be buttoned high enough to cover the chest and the back of the body cannot be exposed.
No sagging pants allowed and pants cannot be worn with the waistband below the hipbone. Underwear cannot be visible at any time. No see-through or mesh clothing that will reveal the body or will reveal underwear.
Oversize clothing or too-tight clothing such as yoga pants, tights, leggings, spandex bicycle/biker shorts, etc. will not be allowed.(Exception: yoga pants, tights, leggings may only be worn under skirt or dress if the skirt or dress is no more than 3 inches above the top of the knee, as stated in rule #1 above,
No headwear (hats, caps, hoods, kerchiefs, curlers, sweatbands, etc.) or sunglasses can be worn inside school buildings. No bandannas.
EXTRA HELP:
Students can make arrangements with me to work Monday through Thursday on an individual basis.
A website is available: where you can find some access to algebra 1 and 2 and geometry books, additional videos and extra assignments to help you succeed. We also have access to tutorials at pearsonsuccessnet.com | 677.169 | 1 |
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Item # 503880
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$72.95
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Description
This book includes many meaningful investigations that use the Casio PRIZM graphing calculator to foster students' algebraic thinking. The investigations are written to both help students experience the value and power of mathematics, and to help them develop a greater understanding of many fundamental concepts in algebra - most notably, the concept of function.
To do so, the investigations use the Casio PRIZM to engage students in modeling data using mathematical functions to represent, analyze, and interpret trends in the real world. For example, students use the PRIZM to model data on the supply and demand of medical doctors in Michigan, the results of an aerial count of elephants in Chad, and the acceleration due to gravity on all of the planets in our solar system. | 677.169 | 1 |
Holt algebra 2 homework help
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Get the Remind app. Students and parents use the Remind app to get updates from their teachers. Visit one of the app stores and download the app to log in and view.
Because of this tremendous increase in the numbers of Asian, Hispanic, and other linguistically and culturally different individuals, school districts can no longer.
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Thousands of users are using our software to conquer their algebra homework. Here are some of their experiences. Pearson Prentice Hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. You have been redirected to Sadlier Connect from one of our product URLs where you used to access additional materials to support your Sadlier program. Math Homework Help. Need math homework help? MathHelp.com's online math lessons are matched to your exact textbook and page number! Get started by. Because of this tremendous increase in the numbers of Asian, Hispanic, and other linguistically and culturally different individuals, school districts can no longer. Standards Documents • High School Mathematics Standards • Coordinate Algebra and Algebra I Crosswalk • Analytic Geometry and Geometry Crosswalk. | 677.169 | 1 |
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Reuse PDF content easily. Change font, size, color, and more. Insert, crop, resize, and delete images. Split a PDF to multiple PDFs. A great cook knows how to take basic ingredients and prepare a delicious meal. In this topic, you will become function-chefs. You will learn how to combine functions with arithmetic operations and how to compose functions. You will also learn how to transform functions in ways that shift, reflect, or stretch their graphs. | 677.169 | 1 |
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The Mathematics K-12 Teachers Need to Know
H. Wu December 19, 2008 Contents Prologue (p. 1) Mathematics for K–12 Teaching (p. 2) Part I: Some Examples (p. 7) Part II: The Mathematics for Teachers of K–8 (p. 22) Prologue In 2001, the Conference Board for Mathematical Sciences published a volume to describe the mathematics that institutions of higher learning should be teaching prospective school teachers ([MET]). It recommends that the mathematical course work for elementary and middle school teachers should be at least 9 and 21 semester-hours, respectively, and for high school teachers it should be the equivalent of a math major plus a 6-hour capstone course connecting college mathematics with school mathematics. The major part of the volume is devoted to a fairly detailed description of the mathematics that elementary, middle, and high school teachers need to know. Given the state of mathematics education in 2008, the recommendation on the course work for teachers by the [MET] volume is very sound, in my opinion. As to the description of the mathematics that teachers need to know, it is such a complicated subject that one would not expect what is in [MET] to be the definitive statement. At the very least, one would want an alternative view from the mathematical perspective. Certain essential features about mathematics tend to be slighted in general education discussions of school mathematics, but
1
here is one occasion when these features need to be brought to the forefront. Mathematical integrity is important where mathematics is concerned, and this is especially true about school mathematics. This paper begins with a general survey of the basic characteristics of mathematics (pp. 2–7). Some examples are then given to illustrate the general discussion (Part I). The bulk of the paper is devoted to a description of the mathematics that teachers of K–8 should know (Part II, pp. 22–69). The omission of what high school teachers should know is partly explained by the fact that a textbook is being written about the mathematics of grades 8-12 for prospective teachers ([Wu 2010]). Mathematics for K–12 Teaching This is the name we give to the body of mathematical knowledge a teacher needs for teaching in schools. At the very least, it includes a slightly more sophisticated version of school mathematics, i.e., all the standard topics in the school mathematics curriculum. In Part II of this article (pp. 22–68), there will be a brief but systematic discussion of what teachers of K–8 need to know about school mathematics. In other words, we will try to quantify as much as possible what this extra bit of "sophistication" is all about. The need for teachers to know school mathematics at a slightly more advanced level than what is found in school textbooks is probably not controversial. After all, if they have to answer students' questions, some of which can be unexpectedly sophisticated, and make up exam problems, a minimal knowledge of school mathematics would not suffice to do either of these activities justice. Perhaps equally non-controversial is the fact that, even within mathematics proper, there is a little bit more beyond the standard skills and concepts in the school curriculum that teachers need to know in order to be successful in the classroom. Teachers have to tell a story when they approach a topic, and the story line, while it is about mathematics, is not part of the normal school mathematics curriculum. They have to motivate their students by explaining why the topic in question is worth learning, and such motivation also does not usually find its way to the school curriculum. To the extent that mathematics is not a collection of tricks to be memorized but a coherent body of knowledge, teachers have to know enough about the discipline to provide continuity from day to day
2
and from lesson to lesson. These connecting currents within mathematics are likewise not part of the school curriculum. Teachers cannot put equal weight on each and every topics in the curriculum because not all topics are created equal; they need to differentiate between the truly basic and the relatively peripheral ones. Teachers cannot make that distinction without an in-depth knowledge of the structure of mathematics. And so on. All this is without a doubt part of the mathematical knowledge that should be part of every teacher's intellectual arsenal, but the various strands of this component of the mathematics for K–12 teaching have so far not been well articulated in the education literature. In the first part of this article, we will attempt such an articulation. To this end, we find it necessary to step back and examine the nature of mathematics education. Beyond the crude realization that mathematics education is about both mathematics and education, we posit that mathematics education is mathematical engineering, in the sense that it is the customization of basic mathematical principles for the consumption of school students ([Wu] 2006). Here we understand "engineering" to be the art or science of customizing scientific theory to meet human needs. Thus chemical engineering is the science of customizing abstract principles in chemistry to help solve day-to-day problems, or electrical engineering is the science of customizing electromagnetic theory to design all the nice gadgets that we have come to consider indispensable. Accepting this proposal that mathematics education is mathematical engineering, we see that school mathematics is the product of the engineering process that converts abstract mathematics into usable lessons in the school classroom, and school mathematics teachers are therefore mathematical engineering technicians in charge of helping the consumers (i.e., the school students) to use this product efficiently and to do repairs when needed. Just as technicians in any kind of engineering must have a "feel" for their profession in order to avert disasters in the myriad unexpected situations they are thrust into, mathematics teachers need to know something about the essence of mathematics in order to successfully carry out their duties in the classroom. To take a simple example, would a teacher be able to tell students that there is no point debating whether a square is a rectangle because it all depends on how one defines a rectangle, and that mathematicians choose to define rectangles to include squares because this inclusion makes more sense in various mathematical settings, such as the discussion of area and volume formulas? This would be a
3
matter of understanding the role of definitions in mathematics. Or, if a teacher y −y finds that the slope of a line L is defined in a textbook to be the ratio x2 −x1 2 1 for two chosen points (x1 , y1 ) and (x2 , y2 ) on L, would she recognize the need to prove to her students that this ratio remains unchanged even when the points (x1 , y1 ) and (x2 , y2 ) are replaced by other points on L? In other words, if (x3 , y3 ) and (x4 , y4 ) are any other pair of points on L, then y2 − y1 y4 − y3 = x2 − x1 x4 − x3 In this case, the teacher has to be alert to the inherent precision of mathematics, so that a definition claiming to express the property of a line should not be formulated solely in terms of two pre-asssigned points on it. It is also about knowing the need to supply reasoning when an assertion is made about the equality of the two ratios. At the moment, our teachers are not given the opportunity to learn about the basic topics of school mathematics, much less the the essence of mathematics ([Ball] 1990; [Wu1999a] and [Wu1999b]; [NRC], pp. 372-378; [MET] 2001, Chapters 1 and 2; [Wu2002a]). For example, the mathematics course requirements of pre-service elementary teachers in most education schools consist of one to two courses (e.g., [NCTQ], p. 25), which are far from adequate for a revisit of the elementary mathematics curriculum in greater depth. One can look at most mathematics textbooks written for elementary teachers, for example, to get an idea of how far we are from providing these teachers with the requisite mathematical knowledge (again, [NCTQ], pp. 35–37). Worse, anecdotal evidence suggests that some mathematics courses that are required may be about "college algebra" or other topics unrelated to the mathematics of elementary school. Along this line, some knowledge of calculus is usually considered a badge of honor among elementary teachers. While more knowledge is always preferable to less, it can be persuasively argued that so long as our elementary teachers don't have a firm grasp of the mathematical topics they have to teach, any knowledge of calculus would be quite beside the point. As for secondary teachers, their required courses are usually taken in the mathematics departments. There is still a general lack of awareness in these departments, however, that the subject of school mathematics is about a body of knowledge distinct from what future mathematicians need for their research ([Wu1999a]), but that it nevertheless deserves their serious attention. At the moment, future secondary school teachers get roughly the
4
same mathematics education as future mathematics graduate students, and the only distinction between these two kinds of education is usually in the form of some pedagogical supplement for the mathematics courses. In general terms, such a glaring lacuna in the professional development of future mathematics teachers is partly due to the failure to recognize that school mathematics is an engineering product and is therefore distinct from the mathematics we teach in standard college mathematics courses. Teaching school mathematics to our prospective teachers requires extra work, and "business as usual" will not get it done. There is another reason. Education itself is beset with many concerns, e.g., equity, pedagogical strategies, cognitive developments, etc. In this mix, schools of education may not give the acquisition of mathematical content knowledge the attention that is its due. And indeed mathematics often gets lost in the shuffle. To further the discussion, more specificity would be necessary. We therefore propose that the following five basic characteristics capture the essence of mathematics that is important for K–12 mathematics teaching: Precision: Mathematical statements are clear and unambiguous. At any moment, it is clear what is known and what is not known. Definitions: They are the bedrock of the mathematical structure. They are the platform that supports reasoning. No definitions, no mathematics. Reasoning: The lifeblood of mathematics. The engine that drives problem solving. Its absence is the root cause of teaching- and learningby-rote. Coherence: Mathematics is a tapestry in which all the concepts and skills are interwoven. it is all of a piece. Purposefulness: Mathematics is goal-oriented, and every concept or skill is there for a purpose. Mathematics is not just fun and games. These characteristics are not independent of each other. For example, without definitions, there would be no reasoning, and without reasoning there would be no coherence to speak of. If they are listed separately, it is only because they provide easy references in any discussion.
5
It may not be out of place to amplify a bit on the characteristic of purposefulness. One reason some students do not feel inspired to learn mathematics is that their lessons are presented to them as something they are supposed to learn, willy-nilly. The fact is that mathematics is a collection of interconnecting chains in which each concept or skill appears as a link in a chain, so that each concept or skill serves the purpose of supporting another one down the line. Students should get to see for themselves that the mathematics curriculum does move forward with a purpose. We can give a first justification of why these five characteristics are important for teaching mathematics in schools. For students who want to be scientists, engineers, or mathematicians, the kind of mathematics they need is the mathematics that respects these basic characteristics. Although this claim is no more than professional judgment at this point, research can clearly be brought to bear on its validity. Accepting this claim for the moment, we see that students are unlikely to learn this kind of mathematics if their teachers don't know it. Apart from the narrow concern for the nation's technological and scientific well-being, we also see from a broader perspective that every student needs to know this kind of mathematics. This is because, if school mathematics education is to live up to its educational potential of providing the best discipline of the mind in the school curriculum, then we would want to expose all students to precise, logical and coherent thinking. This then gives another reason why teachers must know the kind of mathematics that respects these basic characteristics. However, the ultimate justification of why mathematics teachers must know these five characteristics must lie in a demonstration that those who do are better teachers, in the sense that they can make themselves better understood by their students and therefore have a better chance of winning their students' trust. These are testable hypotheses for education research, even if such research is missing at the moment. In the meantime, life goes on. Instead of waiting for research data and doing nothing, we proceed to make a simple argument for the case that teachers should know these basic characteristics, and also give several examples for illustration. The simple argument is that many students are turned off by mathematics because they see it as one giant black box to which even their teachers do not hold the key. Therefore teachers who can make transparent what they are talk6
ing about (cf. definitions and precision), can explain what they ask students to learn (cf. reasoning and coherence), and can explain why students should learn it (cf. purposefulness) have a much better chance of opening up a dialogue with their students and inspiring them to participate in the doing of mathematics. We divide the remaining discussion into two parts. In Part I, we use examples from several standard topics in school mathematics to show how teachers who know the basic characteristics of mathematics are more likely to be able to teach these topics in a meaningful way. Part II highlights the main points in the school curriculum that are often misrepresented in school mathematics. These, therefore, should be the focus of professional development. Part I: Some Examples EXAMPLE 1. Place value. Consider a number such as 7 8 7 5 7 We tell students that the three 7's represent different values as a matter of convention, and yet we expect them to have conceptual understanding of place value. When all is said and done, it is difficult to acquire conceptual understanding of a set of rules. This incongruity between our pedagogical input and the expected outcome causes learning difficulties. Teachers who know the way mathematics is developed through reasoning would look for ways to explain the reason for such a rule. When they do, they will discover that, indeed, the rules of place value are logical consequences of the way we choose to count. This is the decision that we count using only ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Many older numeral systems, for example the Egyptian numeral system, pretty much made up symbols for large numbers as they went along: one symbol for a hundred, another one for a thousand, and yet another one for ten thousand, etc. But by limiting ourselves to the use of ten symbols and no more, we are forced to use more than one position (place) in order to be able to count to large numbers. To illustrate the underlying reasoning and at the same time minimize the enumeration of numbers, we will use three symbols instead of ten: 0, 1, 2. Of
7
course, counting now stops after three steps. To continue, one way is to repeat the three symbols indefinitely: 0 0 0 1 1 1 2 2 2 etc.
This allows us to continue counting all the way to infinity, but the price we pay is that we lose track of where we are in the endless repetitions. In oder to keep track of the repetitions, we label each repetition by a symbol to the left: 00 01 02 10 11 12 20 21 22 Adding one symbol to the left of each group of 0 1 2
The next step is to repeat these 27 numbers indefinitely, and then label each of them by labeling each group of 27 numbers with a 0 and 1 and 2 to the left, thereby obtaining the first 81 (= 27 + 27 + 27) unambiguous numbers in our counting scheme. And so on. This way, students get to see the origin of place value: we use three places only after we have exhausted what we can do with two places. Thus the 2 in 201 stands not for 2, but the third round of repeating the 9 two-digit numbers, i.e., the 2 in 201 signifies that this is a number that comes after the 18th number 200 (in daily life we start counting from 1), and the second and third "digits" 01 signify more precisely that it is in fact the 19th number (9 + 9 + 1). In the same way, if we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and adapt the preceding reasoning, we see that for a number such as 374, the 3 in numbers number, precisely 374 signifies the fourth round of repeating the 100 two-digit and therefore 374 is a number that comes after the 300th 300, while the second and third "digits" 74 signify more that 374 is the 300th-and-70th-and-4th number.
When teachers know the underlying reasoning of place value, they will find a whole host of pedagogical options opened up for them. Instead of simply laying down a set of rules that each place stands for a different value, they can now lead their young charges step-by-step through the counting process and make them see for themselves why each place of a number has a different meaning. Moreover, they can also invite their students to experience the counting process in a different context by using any number of symbols (just as we used three above). In mathematics, it is the case that content knowledge heavily influences pedagogy ([Wu] 2005). EXAMPLE 2. Standard algorithms. The teaching of whole number standard algorithms was a flash point of the Math Wars. As key theorems in the study of whole numbers, there is absolutely no doubt that these algorithms and their explanations should be taught. Nevertheless, their great merit may not always be obvious to elementary students, and the need for teachers to plead their case has thus become a necessity. A teacher must be aware of the two characteristics of definitions and coher9
ence in mathematics in this situation. A minimum requirements for success in teaching these algorithms is to always make explicit the definitions of the four arithmetic operations. For example, whole number addition is by definition continued counting, in the sense that the meaning of 1373+2615 is counting 2615 times beyond the number 1373. A teacher who appreciates the importance of definitions would emphasize this fact by making children add manageable numbers such as 13 + 9 or 39 + 57 by brute force continued counting. When children see what kind of hard work is involved in adding numbers, the addition algorithm comes as a relief because this algorithm allows them to replace the onerous task of continued counting with numbers that may be very large by the continued counting with only single digit numbers. Armed with this realization, they will be more motivated to memorize the addition table for all single digit numbers as well as to learn the addition algorithm. It will also give them incentive to learn the reasoning behind such a marvelous labor-saving device. Multiplication being repeated addition, a simple multiplication such as 48× 27 would require, by definition, the addition of 27 + 27 + · · · + 27 a total of 48 times. In this case, even the addition algorithm would not be a help. Again, a teacher who wants to stress the importance of definitions would, for example, make children get the answer to 7 × 34 by actually performing the repeated addition. Then teaching them the multiplication algorithm and its explanation becomes meaningful. Because this algorithm depends on knowing single-digit multiplications, children get to see why they should memorize the multiplication table. (Purposefulness.) Similar remarks can be made about subtraction and division. The sharp contrast between getting an answer by applying the clumsy definition of each arithmetic operation and by using the relatively simple algorithm serves the purpose of highlighting the virtues of the latter. Therefore a teacher who emphasizes definitions in teaching mathematics would at least have a chance of making a compelling case for the learning of these algorithms, and these algorithms deserve nothing less. More is true. It was mentioned in passing that the addition and multiplication algorithms depend on single-digit computations. It is in fact a unifying theme that the essence of all four standard algorithms is the reduction of any whole number computation to the computation of singledigit numbers. This is a forceful illustration of the coherence of mathematics,
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and a teacher who is alert to this basic characteristic would stress this commonality among the algorithms in her teaching. If a teacher can provide such a conceptual framework for these seemingly disparate algorithmic procedures, she would increase her chances of improving student learning (cf. similar discussions in [Pesek-Kirschner], 2000; [Rittle-Johnson-Alibali], 1999). Incidentally, for an exposition of the division algorithm that brings out the fact that this algorithm is an iteration of single digit computations, see the discussion below in item (A) of Whole numbers, Part II. EXAMPLE 3. Estimation. In recent years, the topic of estimation has become a staple in elementary mathematics instruction. In the form of rounding, it enters most curricula in the second grade. Textbooks routinely ask students to round whole numbers to the nearest one, nearest ten, nearest hundred, etc., without telling them when they should round off or why. A teacher who knows about the purposefulness of mathematics knows that if a skill is worth learning, then it cannot be presented as a meaningless rote exercise. She would introduce in her lessons examples in daily life that naturally call for estimation. For example, would it makes sense to say in a hilly town that the temperature of the day is 73 degrees? (No, because the temperature would depend on the time of the day, the altitude, and the geographic location. Better to round to the nearest 5 or nearest 10. "Approximately 70 degrees" would make more sense.) If Garth lives two blocks from school, would it make sense to say his home is 957 feet away from school? (No, because how to measure the distance? From door to door or from the front of Garth's garden to the front of the school yard? Measured along the edge of the side walk or along the middle of the side walk? Or is the distance measured as the crow flies? Etc.) Better to round to the nearest 50, or at least to the nearest 10. Many other examples such as a city's population or the length of a students' desk (in millimeters) can also be given. When a teacher can present a context and a need for estimation, the rounding of whole numbers becomes a meaningful, and therefore learnable, mathematical skill. Textbooks also present estimation as a tool for checking whether the answer of a computation is reasonable. Here is a typical example. Is 127 + 284 = 411 likely to be correct? One textbook presentation would have students believe that, since rounding to the nearest 100 changes 127 + 284 to 100 + 300, which is 400
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and 400 is close to 411, therefore 411 is a reasonable answer. A teacher who is aware of the need for precision in mathematics would be immediately skeptical about such a presentation. She would ask what is meant by "400 is close to 411". If we change the problem to 147 + 149 = 296, how would this approach to estimation check the reasonableness of the answer? Rounding to the nearest 100 now changes 147 + 149 to 100 + 100, which is 200, should we consider 200 to be "close" to 296? The teacher therefore realizes that even in estimation, there is a need to be precise. She would therefore forsake such a cavalier approach to estimation and teach her students instead about the inevitable errors that come with each estimation. Each rounding to the nearest 100 could bring either an over-estimate or an under-estimate up to 50, and she would teach the notation of ±50 to express this fact. When one adds two such estimations of rounding to the nearest 100, the error could therefore be as high as ±100. And this was exactly what happened with the estimation of 147 + 149: the error of 200 compared with the exact value 296 is 96, which is almost 100. If we use rounding to the nearest 100 to check whether an addition of two 3-digit numbers is reasonable, we must expect an error possibly as high as 100. In this way, she shows her students that the declaration of "closeness" for this way of checking an addition is completely meaningless. She would tell her students that if they really want a good estimate, in the sense of the estimation being within 10 of the true value, they should round to the nearest 10, in which case the previous reasoning yields an error of ±10. By rounding to the nearest 10, the addition 147 + 149 becomes 150 + 150 = 300, and since 296 is within 10 of 300, one may feel somewhat confident of the answer of 296. In this case, teaching the concept of the error of an estimation helps to steer the teacher away from teaching something that is mathematically incorrect. Because the mathematics in school textbooks is often unsatisfactory (compare [Borisovich], or [NMPb], Appendix B of Chapter 3, pp. 3-63 to 3-65), a knowledge about the basic characteristics of mathematics in fact becomes indispensable to the teaching of mathematics. Incidentally, if the consideration of estimation is part of a discussion in a sixth or seventh grade class, then the concepts of "absolute error" and "relative error" should also be taught. EXAMPLE 4. Translations, rotations, reflections. We will refer to these basic concepts in middle school geometry as basic
12
rigid motions. The way these concepts are usually taught, they are treated as an end in itself. Students do exercises to learn about the effects of these basic rigid motions on simple geometric figures. They are also asked to recognize the translation, rotation, and reflection symmetries embedded in patterns and pretty tessellations. This is more or less the extent to which the basic rigid motions are taught in the middle grades. One gets the impression that the basic rigid motions are offered as an aid to art appreciation. A teacher cognizant of the purposefulness of mathematics would try to direct the teaching of these concepts to a mathematical purpose. She would teach the basic rigid motions as the basic building blocks of the fundamental concept of congruence: two geometric figures are by definition congruent if a finite composition of a translation, a rotation, and/or a reflection brings one figure on top of the other. The basic rigid motions are tactile concepts; many hands-on activities using transparencies can be devised to help students get to know them. In school mathematics, congruence is usually defined as "same size and same shape". To the extent that there is no discernible systematic effort in professional development to correct such a lapse of precision, this hazy notion of congruence is what our teachers have be forced to live with.1 By contrast, the definition of congruence in terms of the basic rigid motions is mathematically accurate, is (as noted) tactile, and therefore by comparison learnable. A teacher who knows the basic characteristics of mathematics would have a much better chance of making students understand what congruence is all about. A teacher aware of the coherence of mathematics would also make an effort to direct students' attention to the role played by congruence in other areas of mathematics. She would underscore, for instance, the fact that a basic requirement in the definition of geometric measurements (i.e., length, area, or volume) of geometric figures is that congruent figures have the same geometric measurements. By emphasizing this property of geometric measurement vis-avis congruence, a teacher can greatly clarify many of the usual area or volume computations. (See the discussion in item (D) below on Geometry.) Such considerations enable students to see why they should learn about congruence. (Purposefulness again.) EXAMPLE 5. Similarity.
1
This is another striking example of the absence of mathematical engineering.
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The concept of similarity is usually defined to be "same shape but not necessarily the same size". This phrase carries as much (or if one prefers, as little) information as "same size and same shape". A teacher who values the precision of mathematics would know that this is not a mathematically valid definition of similarity that she can offer to her students. For example, how would "same shape but not necessarily the same size" help decide whether the following two curves are similar, and if so, how?
A correct definition of similarity, one that can be taught to middle school students, turns out to be both elementary and teachable. For convenience, restrict ourselves to the plane. A dilation with center O and scale factor r (r > 0) is the transformation of the plane that leaves O unchanged, but moves any point P distinct from O to the point P so that O, P , P are collinear, P and P are on the same half-line relative to O, and the length of OP is equal to r times the length of OP . Here is an example. Consider the dilation with center O and scale factor 1.8. What does it do to the curve as shown?
What we get is a "profile" of the dilated curve consisting of eleven points.
r r r r r r r rr
r
r
r
r
r
r
r
r r
r
r
r
r
O
q
If we use more points on the original curve and dilate each point the same way, we would get a better approximation to the dilated curve itself. Here is an example with 150 points chosen on the original curve.
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O
q
In the middle school classroom, a teacher can capture students' imagination by telling them that, through the use of dilation, they can expand (scale factor > 1) or contract (scale factor < 1) any geometric figure. In fact, this is the basic principle behind digital photography for expanding or contracting a picture. Both facts are likely to be new to the students. Now the definition of similarity: two geometric figures are similar if a dilation followed by a congruence bring one figure on top of another. Again, such a precise definition makes similarity a tactile and teachable concept. To go back to the two curves at the beginning of this discussion, it turns out that they are similar because, after dilating the right curve by a scale factor of 1.5, we can make it coincide with the curve on the left using the composition of a 90◦ counter-clockwise rotation, a translation, and a reflection across a vertical line. EXAMPLE 6. Fractions, decimals, and percent. Here we focus on the teaching of these topics in grade 5 and up. We do so because this is where the informal knowledge of fractions in the primary grades begins to give way to a formal presentation, and where students' drive to achieve algebra begins to take a serious turn. This is where abstraction becomes absolutely necessary for the first time and, not coincidentally, this is where nonlearning of mathematics begins to take place on a large scale (cf. [Hiebert-Wearne] 1986, [Carpenter-Corbitt et. al] 1981). In the following discussion, we use the term fraction to stand for numbers of the form a , where a and b (b = 0) are whole numbers, and the term decimals b
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to stand for finite decimals. In broad terms, standard instructional materials ask students to believe that a fraction is a piece of pizza, part of a whole, a division, and a ratio; a decimal is a number one writes down by using the concept of extended place value: 43.76 is 4 tens, 3 ones, 7 tenths, and 6 hundredths. a percent is part of a hundred. Teachers who have taught decimals this way are well aware of the elusiveness (to a student) of the concept of extended place value, to the point that students cannot form a concrete image of what a decimal is. One should be able to get research data on this observation directly. And of course, students are also urged to "reason mathematically" using these concepts to solve problems when all they are given is this amorphous mess of information. A teacher who knows the coherence characteristics of mathematics would know that, insofar as fractions, decimals and percents are numbers, the concept of a number should not be presented to students in such a fragmentary manner as suggested by the above sequence of "definitions". For example, the suggested "definition" of a fraction has too many components, and some of them don't even make sense. What is a "ratio"? If students already know what a "ratio" is, would they need a definition of a "fraction"? If a fraction is just a piece of pizza, then how to multiply two pieces of pizza? And these are only two of the most naive concerns. Moreover, since decimals and percent are numbers, it is important that students feel at ease about computing with them. Therefore, if we try to relate this notion of 43.76 to something familiar to students, could it 7 6 be 40 + 3 + 10 + 100 ? If so, isn't a decimal a fraction, and therefore why not say explicitly that a decimal is a fraction obtained by adding the above fractions? By the same token, is "part of a hundred" supposed to mean a fraction whose denominator is 100? If so, why not use this as the definition instead of the imprecise phrase "part of a hundred"? The teacher would recognize the need for a definition of a fraction that is at once precise and correct. One such definition is to say that a fraction is a point on the number line constructed in a precise, prescribed manner, e.g., 2 is the 3 2nd division point to the right of 0 when the segment from 0 to 1 is divided into 3 segments of equal length (see the discussion in item (B) in Fractions, Part II, below; for an extended treatment, see [Wu2002b], and in a slightly different
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form, [Jensen] 2003). Building on this foundation, she can define decimal and percent as special kinds of fractions in the manner described above. After these definitions have been put firmly in place, she would then using reasoning to explore other implications and representations of these concepts. Here are the relevant definitions: a decimal is any fraction with denominator equal to a power of 10, and the decimal point notation is just an abbreviation for the power, 67 e.g. 3.52 and 0.0067 are, by definition, 352 and 104 , respectively, and a 102 percent is a fraction of the form
N 100 N 100 ,
where N is a fraction. (Notation:
is written as N %.)
N 100 above is the fraction obtained 100 A 1 . The division B of two fractions
Note that the
by dividing the fraction N by
the fraction A and B (B = 0) is called in school mathematics a complex fraction, and B will continue to be called the A denominator of the complex fraction B . Therefore a percent is strictly speaking a complex fraction whose denominator is 100.2 What this discussion suggests is that a teacher well aware of the importance of coherence in mathematics would be more likely to find a way to present the concepts of fraction, decimal, and percent from a unified perspective, thereby lighten the cognitive load of students and make these traditionally difficult concepts more transparent and more learnable. Since such a suggestion does not as yet have support by data, it should be a profitable topic for education research. What are the advantages of having a coherent concept of fractions, decimal, and percent? Consider what is supposed to be a difficult problem for sixth and seventh graders: What percent of 76 is 88? How might a teacher who is at ease with the basic characteristics of mathematics handle this? Knowing that percent is a fraction, she would first suggest to her students to look at an easier cognate problem: what fraction of 76 is 88? This
Complex fractions are extremely important for the discussion of ratio and rates in school mathematics and in preparing students for algebra. Their neglect in the school curriculum is inexplicable and inexcusable. See §9 of [Wu2002b].
2
18
problem is, by comparison, more straightforward , namely, if k is the fraction so that k of 76 is 88, then a direct translation of " k of 76 is 88" is3 k × 76 = 88
From this, she gets k = 22 . Now she returns to the original problem: suppose 19 N % of 76 is 88. Since a percent is a fraction, the same reasoning should be used to get N % × 76 = 88
15 Therefore N = 8800 = 115 19 . Thus the answer is 115 15 %. 76 19 We give two more examples. The same teacher would use the definition of a decimal to give a simple explanation of the multiplication algorithm for decimals. For example, the algorithm says that to multiply 2.6 × 0.105,
(α) multiply the corresponding whole numbers 26 × 105, and (β) put the decimal point 4 (= 1+3) places to the left of the last digit of 26×105 because the decimal point in 2.6 (resp., 0.105) is 1 place (resp., 3 places) to the left of the last digit of 26 (resp., 105). She would use this opportunity to illustrate the use of precise definition in mathematical reasoning . She would calculate 2.6 × 0.105 this way: 2.6 × 0.105 = = = 26 105 × 10 103 26 × 105 10 × 103 2730 101+3 (this is (β)) (by definition) (this is (α))
= 0.2730
For the second example, recall that we brought up the concept of extended place value in the usual definition of a decimal. In the education literature, this ad hoc idea has to be accepted on faith for the understanding of a decimal. A teacher
3 This requires a thorough understanding of fraction multiplication. See the discussion in item (B) below in Fractions.
EXAMPLE 7. The equal sign. Education research in algebra sees students' defective understanding of the equal sign as a major reason for their failure to achieve algebra. It is said that students consider the equal sign an announcement of the result of an arithmetic operation rather than as expressing a relation. The conclusion is that the notion of "equal" is complex and difficult for students to comprehend. A teacher who values precision in mathematics would immediately recognize such abuse of the equal sign as a likely result of too much sloppiness in the classroom. She knows how tempting such sloppiness can be. For example, it is so convenient to write "27 divided by 4 has quotient 6 and remainder 3" as 27 ÷ 4 = 6 remainder 3 Here then is a prime example of using the equal sign as "an announcement of the result of an arithmetic operation". But the teacher also recognizes the
20
sad truth that this way of writing division-with-remainder is given in all the standard textbooks as well as in too many professional developmental materials for comfort.4 She knows all too well if teachers are taught the wrong thing, then they will in turn teach their students the wrong thing. If classroom practices and textbooks encourage such sloppiness, students brought up in such an environment would naturally inherit the sloppiness. The so-called misconception of the equal sign is thus likely the inevitable consequence of flawed mathematics instruction that our teachers received from their own teachers and textbooks, which they impart, in turn, on their own students. Again, this is something education research could confirm or refute. From a mathematical perspective, the notion of "equal" is unambiguous and is not difficult to comprehend. The concept of equality is a matter of precise definitions. If teachers can emphasize the importance of definitions, and always define the equal sign in different contexts with precision and care, the chances of students abusing the equal sign would be much smaller. The principal concern for any misunderstanding of the equal sign is therefore something professional development should address. To drive home the point that the concept of equality is a matter of definition, here is the list of the most common definitions of A = B that arise in school mathematics: • A and B are expressions in whole numbers: A and B are verified to be the same number by the process of counting (e.g., A = 2 + 5, B = 4 + 3). If whole numbers are already placed on the number line, then A = B means A and B are the same point on the number line. • A and B are expressions in fractions: same point on number line (e.g., 1 1 1 2 + 3 = 2 − 1 6 ). • A and B are expressions in rational numbers: same point on number line 1 1 (e.g., 3 − 1 = 2 − 2 6 ). 2 • A and B are two sets: A ⊂ B and B ⊂ A. • A and B are two functions: A and B have the same domain of definition, and A(x) = B(x) for all elements x in their common domain.
4
The correct way of expressing the division-with-remainder is of course 27 = (6 × 4) + 3.
21
• A = (a, a ), B = (b, b ) are ordered pairs of numbers: a = b and b = b . (Same for ordered triples of numbers.) • A and B are two abstract polynomials: pairwise equality of the coefficients of the same power of the indeterminate.
Part II: The Mathematics for Teachers of K–8 Let us next examine in some detail what mathematics teachers need to know about school mathematics. Perhaps as a consequence of the hierarchical nature of mathematics, the core content of school mathematics seems to be essentially the same in all the developed countries as far as we know. See [NMPb], Chapter 3, pp. 3-31 to 3-32. True, there are observable minor variations, but these variations all seem to be related to the grade level assigned to each topic or the sequencing of a few of the topics. For example, most Asian countries require calculus in the last year of school, whereas such is rarely the case in the U.S. Or, the U.S. curriculum generally dictate a year of algebra followed by a year of geometry, and then another year of algebra, but such an artificial separation is almost never followed in foreign countries (cf. e.g., [Kodaira], 1996 and 1997). We will have more to say about the later presently. Such differences are, however, insignificant compared with the overall agreement among nations on the core topics in school mathematics before calculus. In terms of grade progression, these core topics are essentially the following: whole number arithmetic, fraction arithmetic, negative numbers, basic geometric concepts, basic geometric mensuration formulas, coordinate system in the plane, linear equations and quadratic equations and their graphs, basic theorems in plane geometry, functions and their graphs, exponential and logarithmic functions, trigonometric functions and their graphs, mathematical induction, binomial theorem. Once we accept that these core topics are what our teachers must teach, the hierarchical nature of mathematics mentioned above dictates, in the main, what teachers must learn and in what order they should learn it. In this sense, teacher's content knowledge is not circumscribed by education research but must also be
22
informed by mathematical judgment. Before we get down to the specifics of the mathematics we want teachers to learn, it behooves us to reflect on the nature of this body of knowledge as it has a bearing on many ongoing discussions about teachers. If we agree that teachers should know a more sophisticated version of school mathematics, the fact that school mathematics is an engineering product (see the discussion of mathematical engineering on page 3) means that what teachers should know must satisfy two seemingly incompatible requirements, namely, (i) it is mathematics that respects the five basic characteristics, and (ii) it is sufficiently close to the established school curriculum so that teachers can make direct use of it in the school classroom without strenuous effort. Let us address specifically the school mathematics of K-8. There is probably no better illustration of the dichotomous nature of school mathematics than the case of fractions, a topic that has been discussed in Example 6 of Part I. From the standpoint of advanced mathematics, the concept of a fraction, and more generally the concept of a rational number, is simplicity itself. Junior level algebra courses in college deal with rational numbers and their arithmetic in less than a week. Given the notorious non-learning of fractions and rational numbers in grades 5–7, it is natural to ask why we don't just use what works in college to teach school students. The simple reason is that the college treatment of fractions requires that we define a fraction as an equivalence class of ordered pairs of integers. It is not just that our average fifth graders, in terms of mathematical maturity, are in no position to work on such an abstract level, but that more pertinently, fifth graders, through their experience, conceive of fractions as parts of a whole, and this conception is worlds apart from ordered pairs of integers or equivalence relations. To facilitate student learning, a theory of fractions for fifth graders would have to take into account such cognitive developments. A course for teachers on fractions, if it is to be useful to them in the school classroom, therefore cannot adopt the abstract approach and ask each prospective teacher to do research of their own in order to bring such abstract knowledge down to the elementary and middle school classsrooms. This kind of research, nontrivial as it is, is best left to professional mathematicians. By the same token,
23
a mathematics course for elementary teachers also must not teach fractions in the usual chaotic and incomprehensible manner (cf. Example 6) and then expect our prospective teacher to miraculously transform such chaos into meaningful lessons in the classroom. Least of all should we expect to achieve improvement in school mathematics education simply by exhorting our teachers to teach for conceptual understanding while continuing to feed these same teachers such chaotic information. We must teach our teachers materials that have gone through the process of mathematical engineering. There have been some attempts to bridge the chasm between the abstract approach and what is useful in the fifth grade classroom. For example, one textbook for professional development defines a fraction a to be the solution b of the equation bx = a, but then it goes on to discuss fractions without once making use of this definition for logical reasoning. In this case, because the "definition" is detached from the logical development, it ceases to be a definition in the mathematical sense. Such a development ill serves both mathematics and education. What was said about the subject of fractions is of course true for most other topics of school mathematics: the concepts of negative numbers, straight line, congruence, similarity, length, area, volume, together with all their associated logical developments. For example, one cannot teach in the school classroom, at any level, the area of a region in the plane as its so-called Lebesgue measure or even the value of an integral. Nor for that matter can one define a line as the graph of a linear equation in two variables (as is done in advance mathematics). What pre-service professional development needs are, as we said, courses in mathematical engineering. In other words, these should be courses which are devoted to mathematics but which have gone through a careful engineering process. Such courses unfortunately have been in short supply in university campuses up to this point, and one can only hope that the situation will change for the better in the near future. Currently, there has been a lot of interest in mathematics teachers' content knowledge, especially in its effect on student achievement. One measure of this content knowledge is the number of mathematics courses teachers have taken. For example, in [Kennedy-Ahn-Choi], it is assumed that "courses in mathematics represent content knowledge". One purpose of the present discussion is to call attention to the fact that, until mathematics departments and schools of educa24
tion take mathematical engineering seriously, mathematics courses are not likely to be very relevant to teachers' ability to teach better in the school classroom, and the number of mathematics courses teachers have taken will continue to be a defective measure of their content knowledge for teaching. But to go back to our task at hand, we now describe, from the standpoint of mathematical engineering, what needs to be taught to teachers of grades K-8, more or less in accordance with the list of core topics enunciated above. It will be clear to one and all that the description itself meticulously observes requirement (ii) above of what teachers should know, namely, it is always close to the average school curriculum. (A) Whole numbers The basis of all mathematics is the whole numbers. In particular, a complete understanding of the whole numbers and its arithmetic operations is the core of the knowledge teachers need in K-3. What is often not recognized is the fact that an adequate understanding of place value, the central concept in discussions of whole numbers, only comes with an understanding of how to count in the Hindu-Arabic numeral system. If teachers have difficulty convincing children that, for example, the 3 in 237 stands not for 3 but for 30, it may be because children only know it as one among hordes of other rules that they have to memorize. Suppose now we explain to teachers the fundamental idea of the Hindu-Arabic numeral system as the use of exactly ten symbols {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} to count indefinitely. Then after the first round of using these symbols to count from 0 up to 9, we would be stuck unless we are allowed to also these ten symbols in a place to its left, as follows. We "recycle" these ten symbols ten times, and each time we systematically place one of the ten symbols {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} to its left so as to keep track of the continued counting. Thus we begin with 00 01 02 03 04 05 06 07 08 09 Then we continue with the same row of numbers but with a 1 placed to the left in place of 0: 10 11 12 13 14 15 16 17 18 19 We continue with the same row of numbers once more but with a 2 instead of 1 placed to the left:
25
20 21 22 23 24 25 26 27 28 29 We continue this way until we get to 90 91 92 93 94 95 96 97 98 99 At this point we can count no more unless we agree to allow the placement of the same ten symbols in another place to the left. We do the same by "recycling" the 100 numbers {00, 01, 02, . . . , 09, 10, 11, . . . . . . , 98, 99} and by pacing each of the ten symbols {1, 2, 3, 4, 5, 6, 7, 8, 9} in succession in the place to its left so as to keep track of the continued counting. Thus, after 000 001 002 · · · 009 010 · · · 097 098 099 we continue with the same row of 100 numbers but with a 1 replacing the 0 on the left: 100 101 102 · · · 110 111 · · · 197 198 199 Then we replace the 1 on the left with a 2: 200 201 202 · · · 210 211 · · · 297 298 299 And so on. We remark that in normal usage, we omit the writing of 0's on the left, so that 001 is just 1, 091 is just 91, etc. As a teacher, the advantage of learning how to count in this fashion is that she now sees the appearance of each digit in each place to the left as a necessity, so that there is no more doubt as to why the 3 in 237 represents 30 and not 3. This is because in the above counting process, we don't get to 37 until we have recycled the 10 symbols {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} a 3rd time, and then count 7 more. With the counting process clearly understood, the teacher would have a far better chance of clearly explaining what place valuer means. More than that, she can also explain, for two whole number m and n, what it means for one to be bigger than the other. Precisely, m is smaller than n, or in symbols, m < n, if m comes before n in the counting process. It then becomes possible to explain why a two digit number is smaller than a three-digit number: a two-digit number is of course a three digit number with a 0 to the left, so in the way counting as done above, those three digit numbers with a leading 0 comes before those with a leading 1, and the latter come before those with a leading 2, etc. In the same way, we see why a number with 12 digits is smaller than one with 14 digits, etc.
26
Without knowing how to count, the fact that a 12 digits number is smaller than a 14 digit number would of course be strictly a matter of faith. A second overriding theme about whole numbers that should be part of the basic knowledge of elementary teachers is the fact that all the standard algorithms (+, −, ×, ÷) reduce whole number computations to single-digit computations. This fact was already discussed in Example 2 of Part I. A recurrent theme of mathematics is in fact to always try to break down complex concepts or skills to simpler ones. The fact that such a simplification is possible in the HinduArabic numeral system is a fantastic achievement. This overriding theme should be emphasized in teaching the standard algorithms because it gives a conceptual framework for the learning of these algorithms. A common failing in the teaching of the standard algorithms is the lack of emphasis on the definitions of the arithmetic operations. (Again, see Example 2 of Part I.) For example, the teaching of the addition algorithm usually goes straight to the column-by-column mechanism and the technique of carrying without a word said about what it means to add two whole numbers. For sure, such teaching necessarily demeans the algorithm to a trick, and nothing more. If we begin by explicitly defining addition as continued counting, so that 1373 + 2615 is counting 2615 times beyond 1373, then students would more likely recognize that such an addition problem is no easy task. When this is understood, then the extremely simple procedure of obtaining the answer by doing four single-digit additions, 3 + 5, 7 + 1, 3 + 6, and 1 + 2, becomes truly impressive. All teachers should be able to convey this sense of wonderment to their students, and this would not happen if teachers are not taught this knowledge. Once the teachers are secure in this knowledge, then they would recognize the extra bit of work to cope with the phenomenon of carrying in the addition algorithm is just that, an extra bit of work. The same remark applies to the subtraction algorithm and trading. Because the long division algorithm is the most challenging of the four standard algorithms, a few comments may be in order. First of all, the name of the "long division algorithm" is misleading: it is not about division per se but about division-with-remainder; the latter is not a "division" in the usual mathematical sense. In the context of school mathematics as of 2008, division-with-remainder needs careful explanation for the reason that it is so carelessly taught in general as to not even give a proper definition of the concept of the remainder. But as is
27
well-known, division-with-remainder is an important topic because it underlies the Euclidean algorithm (see below) and the division algorithm of polynomials. But to return to the long division algorithm, it is an iteration of divisionswith-remainder. Here is an example to clarify this comment. Consider the division of 586 by 3. At the outset, it should be made clear that what the long division algorithm is about: it is an algorithm to compute, digit by digit, the quotient and the remainder of the division-with-remainder of 586 by 3 which, when expressed correctly, states: 586 = (195 × 3) + 1 School mathematics has a long tradition of expressing this fact as 583 ÷ 3 = 195 R1. This is an outright abuse of the equal sign, which contributes to much confusion in education research and should be studiously avoided (see Example 7 of Part I). Now, the usual way to express the long division algorithm is the following:
3 ) 1 5 3 2 2 9 8 8 7 1 1 5 6
6 5 1
It is not difficult to see that the algorithm, in this special case, is completely captured by the following three simpler divisions-with-remainder, which will be referred to as the procedural description of the algorithm. Observe that the quotient can be read off, digit-by-digit, vertically from the first entries on the right sides and that the remainder is the last number in the last line: 5 = ( 1 × 3) + 2 28 = ( 9 × 3) + 1 16 = ( 5 × 3) + 1 The mechanism for going from one division-with-remainder in this array to the next is the following: the first division-with-remainder takes the left digit of the dividend (586) as its own dividend, and in general, the dividend of each succeeding division-with-remainder is obtained by taking the remainder of the preceding
28
one, multiply it by 10, and add to it the next digit in the original dividend (586 in this case). It is now a simple exercise to make use of the procedural description together with the expanded form of 586 as 586 = (5 × 102 ) + (8 × 10) + 6 to derive the desired conclusion that, in fact, 586 = (195 × 3) + 1. Insofar as this explanation does not depend on the specific numbers 586 and 3, it gives a general understanding of why the long division algorithm always yields the correct division-with-remainder for any two whole numbers. Teachers should get to understand this beautiful algorithm in such depth before good teaching can take place in the school classroom. The basic technique of division-with-remainder has multiple implications in the school classroom. The first are the various divisibility rules, such as a whole number n is divisible by 3 if and only if the whole number obtained by adding the digits of n is divisible by 3. Emphasize that all such divisibility rules are nothing more than a consequence of the behavior of a power of 10 when it is divisible by a single digit number (such as 3). (Incidentally, the divisibility rule for 7 is so complicated that it does not deserve to be taught in the professional development of elementary teachers.) Because these divisibility rules are usually taught as unexplained tricks in the elementary classroom, their explanations are overdue. A more substantial application of division-with-remainder is the Euclidean algorithm, which is an algorithm for finding the GCD (greatest common divisor) of two whole numbers. (It is not necessary, but it is easier if the concept of an integer is available at this juncture.) In the process of developing this algorithm, the following basic fact will be uncovered: the GCD k of two whole numbers a and b can be expressed as: k = ma + nb for some integers m and n. From this, it follows that if a prime number p divides a product of whole numbers ab, and p does not divide a, then p must divide b. The Fundamental Theorem of Arithmetic is now a simple consequence. Students in grades 5-7 may not fully grasp the significance of the uniqueness statement, but elementary teachers must make an effort to come to an understanding of this fact. The concept of uniqueness, while subtle (Euclid did not have it, for example), is of fundamental importance in modern mathematics, but it is rarely taught in K–12. If elementary teachers can begin to informally teach this idea,
29
and middle and high school teachers can continue to keep this idea alive in classroom discussions, all students would benefit from such instruction. In recent years, the subject of estimation has been emphasized in K–6, and rightly so, but what has found its way into textbooks on the subject tends to misrepresent the reason for this emphasis. (See Example 3 of Part I.) Let us begin with an enumeration of some of the troubling issues. The first one is that it is difficult to get a correct description of the rounding of numbers to the nearest 10, nearest 100, nearest 1000, etc. In standard textbooks, students are usually taught to round a whole number n to the nearest 10 by the following algorithm: if the ones digit of n is ≤ 4, change it to 0 and leave the other digits unchanged, but if the ones digit is b ≥ 5, then change it to 0 but also increase the tens digit by 1 and leave other digits unchanged. This is correct in most cases, but collapses completely in the case of a number such as 12996. A correct formulation of rounding a whole number n to the nearest 10 is the following: Write n as N + n, where n is the single-digit number equal to the ones digit of n (and hence N is the whole number obtained from n by replacing its ones digit with 0). Then rounding n to the nearest ten yields the number which is equal to N if n < 5, and equal to N + 10 if n ≥ 5. One can give an analogous formulation for rounding to other powers of 10. A more fundamental issue is that estimation is taught as a rote activity with no thought given to convincing students that this is something worth learning. Students are not told why sometimes they should estimate (e.g., uncertainties in a measurement), under what circumstance an approximate answer is all that makes sense (e.g., built-in imprecision in the concept, such as distance from house to school or temperature of the day), or under what circumstance an estimation becomes an aid to achieving precision (e.g., the process of carrying out the long division algorithm). An even more serious concern is that students are not alerted to the need of always finding out about the error that comes with each estimation. In grades up to five (approximately), it is understood that only the concept of absolute error can be discussed, but starting with roughly grade six, students should be taught to routinely estimate the percentage error. Professional development would do well to take these potential pitfalls into account and provide teachers with the kind of instruction that would enable them
30
to avoid such pitfalls. (B) Fractions Fractions are positive rational numbers for this discussion. Compare [Wu 2008], especially with regard to the research literature. We will address fractions in this section, and negative rational numbers in the next. Before we begin the detailed discussion of teachers' knowledge of fractions, it should be stated up front that the kind of knowledge discussed below is essentially one that can be used in the classroom of grades 5–7 without drastic changes. Where then does this discussion leave the primary teachers? We believe that all elementary teachers, including primary teachers, should acquire such knowledge about fractions. The most obvious reason is that no elementary teacher can guarantee that they will teach only the primary grades the rest of their lives. The real reason is, however, the fact that what a teacher teaches in the primary grades may be simple, but it should still be a simplified version of correct mathematics. For example, a teacher familiar with a logical development of fractions would recognize the futility of relying exclusively on cutting pizzas in order to teach fractions. Such a teacher would be more likely to introduce the number line as early as possible. If a teacher knows how fractions can be developed in a way that is consonant with the basic characteristics of mathematics, then she would be immeasurably better equipped to provide primary students with the mathematical foundation they need in the later grades. Traditionally, students' failure to learn fractions is explained by the disconnection in their understanding of the conceptual complexity of fractions (e.g., [Behr-Lesh-Post-Silver] 1983, [Bezuk-Bieck] 1993). Our teachers are therefore exhorted to develop a strong number sense about fractions and to develop an ability to think about fractions "in other ways" beyond part-whole. A recurrent theme in the education literature is that, to achieve flexibility in working with rational numbers, one must acquire a solid understanding of the different representations for fractions, decimals, and percents. Another theme, equally prized, is that a deep understanding of rational numbers should be developed through experiences with a variety of models, such as fraction strips, number lines, 10×10 grids, and area models. There are two things fundamentally wrong with such a view of fractions. The
31
first is that, if students are not told what a fraction is, any talk about their "different representations" of a fraction would be akin to talking about the spots on the skin of a unicorn. It is appealing, but it is educationally unsound. (See Example 6 of Part I.) The second thing wrong is that, up to this point, there has been hardly any mathematically correct presentation of the subject of fractions in the school classroom or pre-service professional development. Against this background, to talk about developing a "deep understanding" of fractions through hands-on experiences is therefore to concede that an incoherent presentation of fractions is a law of nature, so that we must settle for ultra-mathematical methods for the learning of fractions. Although such a view happens to be consistent with one commonly held in mathematics education research (cf. papers from the Rational Number Project, e.g., [Behr-Harel-Post-Lesh] 1992; also [Kieren] 1976 and [Vergnaud] 1983), it is not a mathematically acceptable view of a topic in mathematics. Teachers' understanding of fractions as a mathematical concept will not improve until we can provide them with a mathematical framework in which all these "representations" emerge as logical consequences of a clearly enunciated conception (definition) of a fraction. (Compare the discussion of coherence in Examples 2, 4, and 6 of Part I.) We want teachers to see that the subject of fractions is one that is infused with the aforementioned five characteristic properties of mathematics. Because the tradition of teaching by rote in fractions is so entrenched, unless we can make our teachers buy into the idea that the subject of fractions is logical rather than whimsical, there would be little hope that their students would perceive fractions as a learnable subject. We need to approach fractions from a viewpoint that is consonant with requirements (i) and (ii) above. In other words, we need an approach that is rooted in mathematical engineering. Such an approach has been available for some time now ([Wu 2002]; for a similar but slightly different one, see [Jensen 2003]). Regardless of how soon school textbooks will teach fractions as part of mathematics (rather than as part experimental science and part royal decree), there is an urgent need for our teachers to learn a mathematically valid presentation of fractions. Such a presentation begins with a definition of fractions as points on the
32
number line constructed in a specific manner and, on this basis, provides all the properties we expect of fractions with mathematical explanations. This is not the place to give a detailed treatment or even a complete summary of such a presentation. We can, however, try to give a flavor of what this presentation tries to accomplish. For example, let us see what is a reasonable definition of the fractions with denominator equal to 3. We first fix some terminology. If a and b are two points on the number line, with a to the left of b, we denote the segment from a to b by [a, b]. The points a and b are called the endpoints of [a, b]. The special case of the segment [0, 1] occupies a distinguished position in the study of fractions; it is called the unit segment and its length is, intuitively, our "whole". The point 1 is called the unit. Since the fraction 1 is one-third of 3 the whole, we see from the picture of the number line below that the length of any of the three smaller segments of equal length between 0 and 1 qualifies as 1 . 3 However, the right endpoint of the thickened segment is sufficient, in an intuitive sense, to indicate the length of this thickened segment, so this right end-point will be chosen as the representative of 1 . 3 0
1 3
1
2
3
If we divide, not just [0, 1], but every segment between two consecutive whole numbers — [0, 1], [1, 2], [2, 3], [3, 4], etc. — into three equal parts, then these division points together with the whole numbers form an infinite sequence of equi-spaced points, to be called the sequence of thirds. The point in this 1 sequence to the right of 3 will be called 2 , the third 3 , etc. 3 3 0 1 2 3 4
0 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 3 11 3 12 3
Observes that each point in this sequence gives the length of the segment from 0 to the point itself. For example, 10 is the length of the segment [0, 10 ], as the 3 3 latter is 10 times the length of [0, 1 ] (just count!). 3 0 1 2 3
10 3
33
We have now used intuitive reasoning to locate the fractions with denominator equal to 3 on the number line. In a formal mathematical introduction to fractions, we would therefore first create the sequence of thirds in exactly the same way, and then define the fractions with denominator equal to 3 to be exactly these points. In other words, as far as mathematics is concerned, the fraction 10 (for example) is just the tenth 3 point in this sequence to the right of 0, no more and no less. If we want to say anything about this fraction, we must start with the fact that it is the tenth point in the sequence of thirds. In like manner, the fractions with denominator equal to n are by definition the points in the following sequence: we divide each of [0, 1], [1, 2], [2, 3], . . . , into n equal parts, then these division points together with the whole numbers form the sought-for infinite sequence of equi-spaced points. This sequence is called the sequence of nths. The fraction m is then the m-th point to the right of 0 n in this infinite sequence. Here we can point to an immediate advantage of having such a precise definition of a fraction. Given two fractions A and B, we define A < B, and say A is less than (or smaller than) B if A is to the left of B on the number line. Why this is significant is that, in the traditional presentation of fractions, there is no definition of what it means for one fraction to be smaller than another. Rather, students are told to do something first (e.g., get a common denominator for both fractions) and then decide after the fact that one is smaller than the other. In the following, the whole number n in a fraction symbol m will be automatn ically assumed to be nonzero. A special class of fractions are those whose denominators are all positive powers of 10, e.g., 1489 24 58900 , , . 102 105 104 These are called decimal fractions, but they are usually abbreviated to 14.89, 0.00024, 5.8900
respectively. The rationale of the notation is clear: the number of digits to the right of the so-called decimal point keeps track of the power of 10 in
34
the respective denominators, 2 in 14.89, 5 in 0.00024, and 4 in 5.8900. In this notation, these numbers are called finite or terminating decimals. In context, we usually omit any mention of "finite" or "terminating" and just say "decimals" if there is no danger of confusion. One would like to think that 5.8900 is the same as 5.89, as every school student is taught about this at the outset, but we have already agree by definition that 5.8900 is 58900 whereas 5.89 is (again by 104
89 definition) 102 . How do we know that they are equal?
58900 589 = 2 104 10 We must give a proof! It turns out to be more enlightening to first prove a general fact which is fundamental to the whole subject of fractions. This is the theorem known in school mathematics as "equivalent fractions". First, we define two fractions to be equal if they are the same point on the number line. The Theorem on Equivalent fractions is the statement that given two fractions m and k , if there is a whole number j so that k = jm and = jn, then n
m n
= k.
It is common to state this theorem in the following form: for all whole numbers j, m, and n (so that n = 0 and j = 0), m jm = n jn This fact can be simply proved with the definition of a fraction available. We give the reasoning for the special case 4 5×4 = 3 5×3 but this reasoning will be seen to hold in general. First locate line: 0
T T T 4 3
on the number
1
T
4 3 T T
35
We divide each of the segments between consecutive points in the sequence of thirds into 5 equal parts. Then each of the segments [0, 1], [1, 2], [2, 3], . . . is now divided into 15 equal parts and, in an obvious way, we have obtained the sequence of fifteenths on the number line: 0
T T T
1
T
4 3 T T
The point 4 , being the 4-th point in the sequence of thirds, is now the 20-th 3 20 point in the sequence of fifteenths. The latter is by definition the fraction 15 , 5×4 i.e., 5×4 . Thus 4 = 5×3 . 5×3 3 Observe that without a precise definition of a fraction, it would be difficult to make sense of the statement of equivalent fractions for arbitrary j, m and n. The first application of the theorem on equivalent fractions is to bring closure to the discussion about the decimal 5.8900. Recall that we had, by definition, 58900 = 5.8900 104 We now show that 5.8900 = 5.89 and, more generally, one can add or delete zeros to the right end of the decimal point without changing the decimal. Indeed, 58900 589 × 102 589 5.8900 = = 2 = 2 = 5.89, 104 10 × 102 10 where the middle equality makes use of equivalent fractions. The reasoning is of course valid in general, e.g., 12.7 = 127 127 × 104 1270000 = = = 12.70000 10 10 × 104 105
It is commonly stated that a conceptual understanding of fractions should include the fact that a fraction is both parts-of-a-whole and a division. We hasten to point out that this statement is meaningless as a statement in mathematics.
36
Indeed, one is supposed to understand, for example, that the fraction 5 is not 7 only 5 parts when the whole is divided into 7 equal parts, but also "5 divided by 7". The first questionable aspect of this statement is that if 5 does mean "5 7 divided by 7", then one must be able to give a reason. In other words, there is a theorem to prove, although none is ever offered. A second questionable aspect is that, among whole numbers, there is no division such as "5 ÷ 7", only divisions of the form 12 ÷ 4, 25 ÷ 5, 48 ÷ 6, or in general, a ÷ b when a is a multiple of b. Therefore just to make sense of m as "m ÷ n", we must first precisely define n for any two whole numbers m, n (n = 0), that m ÷ n is the length of one part when a segment of length m is partitioned5 into n equal parts. Now we are at least in a position to make an assertion that is mathematically meaningful, namely, we assert the equality of two numbers, m and m ÷ n:6 n m =m÷n n This is the correct meaning of the so-called division interpretation of a fraction. And, of course, we still need to give a proof! The advantage of having done all this work to clarify this statement is that we see more clearly how to prove it. To divide [0, m] into n equal parts, we express m = m as 1 nm n
1 using equivalent fractons. That is, nm copies of n , which is equivalent to n copies of m . So one part out of these n equal parts is just m . n n
We note once again that such a precise explanation could be given only because we have a precise definition of a fraction. No professional development for elementary teachers can afford to avoid a discussion of whether a teacher can insist on always reducing a fraction to lowest terms. Implicit in this stance is the statement that
To avoid the possibly confusing appearance of the word "divide" at this juncture, we have intentionally used "partition" instead. 6 Notice how careful we are in using the equal sign! Compare Example 7 in Part I.
5
37
every fraction is equal to a unique fraction (one and only one fraction) in lowest terms. First of all, we must recognize the fact that it is quite nontrivial to prove this statement. This is where the Euclidean algorithm comes in. While a proof should be given, it cannot be given in grade 5 or even in grade 6 in most schools because of the mathematical sophistication involved. In addition, teachers should also 4 know that, a fraction such as 12 is every bit as good as 3 , so that the insistence 9 that 4 be use rather than 12 must be recognized as a preference but not a math3 9 ematical necessity. Finally, it is sometimes not immediately obvious whether a fraction is in lowest terms or not, e.g., 68 . (It is not.) For all these reasons, a 51 more flexible attitude towards unreduced fractions would consequently make for a better mathematics education for students. One would go on to define the addition and subtraction of fractions, the multiplication of fractions, and the division of fractions. Here, we want teachers to appreciate the coherence of mathematics by exhibiting the fundamental similarity between the arithmetic operations on fractions and those on whole numbers. See [Wu 2001]. Teachers need to appreciate the fact that fractions are not "another kind of numbers". To define the addition of two fractions, we first consider how we add whole numbers when whole numbers are considered as points on the number line. Take for example, the addition of 4 to 7. In terms of the number line, this is just the total length of the two segments joined together end-to-end, one of length 4 and the other of length 7, which is of course 11, as shown. 0 4 11 4 7 We call this process the concatenation of the two segments. Imitating this
k process, we define, given fractions k and m , their sum + m by n n k m + = the length of two concatenated segments, one n of length k , followed by one of length m n
k k ' m n
+
38
m n
E
Then one proves the addition formula for any two fractions k +
m
k
and
m n:
m kn + m = n n
For the subtraction − n to make sense, we first have to make sure that k > m . Once done, the subtraction is then defined to be the length of the n remaining segment when a segment of length m is taken away from one end of a n segment of length k . 0
k
k
−
m n m n
k
A Next, given two fractions A and B, we will define A×B and B . We first define k of a number x to be the number which is the length of the concatenation of k parts when the segment [0, x] of length x is partitioned into parts of equal
Observe that when = 1, k = k, so that (if m is a whole number) "k of m" is according to the above definition exactly the length of k copies of [0, m], i.e., k × m is km, which is the definition of multiplication among whole numbers. It should mentioned that the product formula leads to the second, and equally important meaning of fraction multiplication: A Finally, given fractions A and B (B = 0), the division or quotient B is by definition the fraction C so that A = CB. If this doesn't sound familiar, consider the division of whole numbers such as 36 . We tell children that 36 = 4 9 9 because 36 = 4 × 9. Now if replace 36 by A, 9 by B, and 4 by C, then we would A get exactly the definition of B . (We should add that, in the preceding definition A of B , the existence of a unique fraction C that satisfies A = CB must be proved.) The classical rule of invert-and-multiply now becomes a theorem.
39
Once the concept of division is available, we can introduce the important A concept of a complex fraction, i.e., the division B , where A and B are fractions (B = 0). Now a complex fraction is just a fraction, so why single it out for discussion? To see this, consider the sum of the two complex fractions such as 2.8
4 5
+
12 7
2.5
We know how to do the addition: express each complex fraction as a fraction by the invert-and-multiply rule, 2.8
4 5
However, suppose we make believe that the complex fractions are just ordinary fractions and we add them as we would ordinary fractions. Then the addition formula for fractions yields the same answer: 2.8
4 5
(2.8 × 2.5) + ( 4 × 5 = + 4 2.5 × 2.5 5
12 7
12 7)
7 + 48 35 = 7 + 24 = 293 = 2 2 35 70
Two thoughts immediately come to mind. One is that although the second strategy is blatantly illegal at this point (the addition formula has been proved only for ordinary fractions), it nevertheless gives the correct answer. Is it just luck? We will show that it is not. A second thought is that, since the first strategy always works, why bother with the second one? The superficial reason is that because the second strategy uses the same mechanical procedure for both ordinary fractions and complex fractions, it has the advantage of saving some wear-and-tear of the brain. But the real reason is that when we come to the manipulation of rational expressions in algebra, we will be forced to use the second strategy and will no longer have a choice.
40
This leads us to the arithmetic of complex fractions: can we add, subtract, multiply, and divide them as if they were ordinary fractions (see above)? The answer is yes. Textbooks and the education literature take this fact for granted and make use of it without a word of explanation. They do not consider it necessary to point out that such an extension of the arithmetic of fractions to complex fractions has taken place and, even more importantly, that it is correct. What we wish to affirm is that, indeed, every single one of the computational formulas involving fractions can be proved to be valid for complex fractions (though the proofs are mechanical and not interesting). Our main point is, however, that it is bad policy for school mathematics to be so cavalier about this generalization — from fractions to complex fractions — and we must at least get our teachers to understand why this is a bad policy. Students should learn not to overstep the bounds of what they know. If they want to claim more than they know, they should be immediately aware of the need to prove it. In this instance, it is a matter of luck that the extrapolation from fractions to complex fractions turns out not to cause any problems. One cannot, however, expect this kind of luck to persist. For example, among fractions, it is true that for any fraction A, the fact that B ≥ C implies that AB ≥ AC. Students who have formed the habit of claiming more than they know would assume, when they come to rational numbers (i.e., positive and negative fractions, see item (C), Rational numbers, below), that for any rational numbers A, B, C, B ≥ C also implies AB ≥ AC. This would be a mistake, because while 3 ≥ 2, it is false that (−4)3 ≥ (−4)2 because (−4)3 = −12, which is less than (−4)2 = −8. If we do not want students to fall into this bad habit, then we are obligated to make sure that our teachers do not form such bad habits in the first place. Complex fractions are critical to the study of fractions and should be singled out and systematically taught to teachers. To demonstrate their importance, let us introduce the concepts of percent, ratio, and rate in general. A percent is a complex fraction whose denominator is 100. By tradition, a N N percent 100 , where N is a fraction, is often written as N %. By regarding 100 as an ordinary fraction, we see that the usual statement N % of a number
m n
definition of percent available, all questions about percent can be routinely computed. See Example 6 in Part I. Next, given two fractions A and B (B = 0), both referring to the same unit (i.e., they are points on the same number line), the ratio of A to B, sometimes A denoted by A : B, is by definition the complex fraction B . In connection with ratio, there is a common expression that needs to be made explicit. To say that the ratio of boys to girls in a classroom is 3 to 2 is to say, by convention, that if B (resp., G) is the number of boys (resp., girls) in the classroom, then 3 the ratio of B to G is 2 . When A and B are whole numbers, we want to show that this definition of the ratio 5 : 7 has the same intuitive meaning as "5 parts to 7 parts". Indeed, 5 : 7 is by definition the fraction 5 which, by the definition of a fraction, is the 7 5th division point when the unit segment [0, 1] is divided into 7 equal parts:
c c 6 7
0
1 7
2 7
3 7
4 7
5 7
1
5 1 We therefore see that 5 : 7 (i.e., 7 ) is 5 parts (a part being 7 ) compared with 7 parts. The significance of this definition of the ratio of a fraction A to a fraction B is that, by first establishing the meaning of a fraction as a point on the number line, we show that when A and B are both whole numbers, the meaning of A : B A is exactly the fraction B . As is well-known, one of the traditional definitions of A a fraction B (A, B are whole numbers, B = 0) is that it is the ratio of A to B. What we have done is therefore to clarify the relationship between these two concepts by turning the table: we define fractions first and then define ratio in terms of a fraction.
In school mathematics, the most substantial application of the concept of division is to problems related to rate, or more precisely, constant rate. The precise definition of the general concept of "rate" requires more advanced mathematics, and in any case, it is irrelevant in school mathematics whether we know what a rate is or not. What is important is to know the precise meaning of "constant rate" in specific situations, and some of the most common ones will now
42
be described. The most intuitive among the various kinds of rate is speed. A motion is of constant speed v (v being a fixed number) if the distance traveled, d, from time 0 to any time t is d = vt. Equivalently, in terms of the concept of division, a motion is of constant speed if there is a fixed number v, so that for any positive number t, the distance d (feet, miles, etc.) traveled in any time interval of length t (seconds, minutes, etc.) satisfies d =v t Notice that d is a complex fraction, from which, one can infer that most of the t computations in speed problems involve the arithmetic of complex fractions. What is noteworthy about the preceding equation is the fact that we are dividing two numbers, d and t, ostensibly from different number lines. In greater detail, d is a number on the number line where 1 is the chosen unit of length (foot, mile, etc.) while t is on the number line whose unit 1 is the chosen unit of time (second, minute, etc.). What we have done, at least implicitly, is to identify the two units of length and time, so that d and t are now points on the same number line. If this sounds strange, it could only be because it is rarely explicitly pointed out, although it is done all the time. For example, suppose a rectangle has area 48 ft2 and one side is 8 ft. The length of the other side is then 48 = 6 ft. 8 Here the division makes sense only because we have identified the unit ft2 with the unit of length, one foot. In any case, it is the need of identifying two number lines that distinguishes rate from ratio. In the language of school mathematics, speed is the "rate" at which the work of going from one place to another is done. Other standard "rate" problems which deserve to be mentioned are the following. One of them is painting (the exterior of) a house. The rate there would be the number of square feet painted per day or per hour. A second one is mowing a lawn. The rate in question would be the number of square feet mowed per hour or per minute. A third is the work done by water flowing out of a faucet, and the rate is the number of gallons of water coming out per minute or per second. In each case, the concept of constant rate can be defined in a manner that is identical to the case of constant speed. For example, a constant rate of lawn-mowing would mean: there is a constant r (with unit equal to square-feet-per-hour) so that if A is the total area mowed after T hours, then A = rT no matter what T is.
43
Equivalently, the lawn-mowing is of constant rate if there is a fixed number r so that the number of square feet A mowed in T hours satisfies A =r T no matter what T is. Without knowing the precise meanings of division and multiplication among fractions, it would be impossible to detect the fact that all these constant rate problems are identical problems. For example, assuming constant rate in each situation, the problem of "if I walk 287 meters in 9 minutes, how many meters do I walk in 7 minutes?" is identical to "if I mow 287 square meters of lawn in 9 minutes, how many square meters do I mow in 7 minutes?". This is one argument for emphasizing the importance of definitions. Finally, we take up the topic of converting a fraction to a finite or infinite decimal. We take this up last because of its deceptive subtlety. Consider first the case of those fractions which are equal to a finite decimal. (Teachers should learn how to prove the theorem that a reduced fraction is equal to a finite decimal if and only if its denominator only has 2 or 5 as its prime factors. This proof needs the Fundamental Theorem of Arithmetic.) Let us prove, for example, that 3 = 0.375 8 By itself, this equality is unremarkable. Indeed, the definition of 0.375 is that, by equivalent fractions, we get 0.375 = 375 3 × 125 3 = = 1000 8 × 125 8
375 1000 ,
so
However, the algorithm that converts a fraction to decimals asserts that one 3 obtains the decimal 0.375 from the fraction 8 by doing the long division of 3 × 105 (or 3×10n for any large n) by 8 and then placing the decimal point in the quotient in some prescribed way. Thus what is at issue here is not so much that the two numbers 3 and 0.375 are equal,7 but that the method of long division of 300000 8
But note the importance in this context of having precise definitions of both 3 , a fraction, and 0.375, a 8 decimal, as numbers. Anything less (such as only knowing a fraction as a piece of pizza) and this equality wouldn't even make sense.
7
44
by 8 would yield the correct answer. This can be done simply as follows: 3 1 3 × 105 = 5× 8 10 8 By the long division algorithm, 3×10 = 37500. Therefore, using the definition of 8 a decimal, we have: 37500 1 3 = 5 × 37500 = = 0.37500 = 0.375 8 10 105 Clearly, we would obtain the same answer if 105 is replaced by 103 , or any power of 10 greater than 3. In general, we just try to multiply the numerator and denominator of the fraction under consideration by a large power of 10, where "large" means "large enough to see the decimal terminate in 0's". The same reasoning is applicable to all other cases. In general, a fraction is equal to an infinite repeating decimal. For example, 3 = 0.428571428571428571428571428571 . . . 7 The task of proving this is not so simple. It involves (i) making sense of an infinite decimal as a point on the number line, (ii) showing that through the process of 3 long division the fraction 7 is equal to the above infinite decimal, and (iii) proving that the infinite decimal is necessarily repeating. In school textbooks, basically none of these three steps is proved (though there may be some halfhearted attempt at explaining (iii)), which is understandable considering the advance nature of the mathematics involved. Professional development materials usually concentrate on explaining (iii) but completely ignore (ii) and (i). It would be a good idea to tread lightly on (i), at least for elementary teacher, but (ii) should be taken up seriously in professional development. One shows directly using the mechanism of the long division algorithm (see part (A) on Whole numbers) that 4 2 8 5 7 1 3 = + 2+ 3+ 4+ 5+ 6+ 7 10 10 10 10 10 10 1 4 2 8 5 7 1 + 2 + 3 + 4 + 5 + 6 + ··· 106 10 10 10 10 10 10
3 This then is the meaning of 7 = 0.428571428571 . . . on one level. On a deeper level, we need to prove the convergence of the infinite series.
5
45
(C) Rational numbers Like the teaching of fractions, the teaching of rational numbers (positive and negative fractions) is usually nothing more than the presentation of a collection of rules to be memorized, with an occasional pseudo-explanation thrown in (such as the many analogies purporting to show why negative × negative = positive). Rational numbers present a higher level of abstraction than fractions, and can be understood only if the abstract laws of operations, especially the distributive law for both positive and negative fractions, are taken seriously. Few of our teachers get this message in their college courses. This is a subject littered with plausible statements promoted unceremoniously x as truths without explanations, e.g., the statement that −x = −y = − x for all y y rational numbers (not just whole numbers) x and y. Nowhere is it more important that we carefully attend to the clarity of the definition of each concept and the proof of every assertion. With a number understood to be a point on the number line, we now look at all the numbers as a whole. Take any point p on the number line which is not equal to 0; such a p could be on either side of 0 and, in particular, it does not have to be a fraction. Denote its mirror reflection on the opposite side of 0 by p∗ , i.e., p and p∗ are equidistant from 0 and are on opposite sides of 0. We simply call p∗ the mirror reflection of p. If p = 0, we define 0∗ = 0 Then for any points p, it is clear that p∗∗ = p This is nothing but a succinct way of expressing the fact that reflecting a nonzero point across 0 twice in succession brings it back to itself (if p = 0, of course 0∗∗ = 0). Here are two examples of mirror reflections:
q
p∗
0
p
q∗
Because the fractions are to the right of 0, the numbers such as 1∗ , 2∗ , or ( 9 )∗ are to the left of 0. The set of all the fractions and their mirror reflections 5
46
with respect to 0, i.e., the numbers m and ( k )∗ for all whole numbers k, l, n l m, n (l = 0, n = 0), is called the rational numbers. Recall that the whole numbers are a sub-set of the fractions. The set of whole numbers and their mirror reflections, . . . 3∗ , 2∗ , 1∗ , 0, 1, 2, 3, . . . is called the integers. We therefore have: whole numbers ⊂ integers ⊂ rational numbers We now extend the order among numbers from fractions to all numbers: for any x, y on the number line, x < y means that x is to the left of y. An equivalent notation is y > x. x y
Numbers which are to the right of 0 (thus those x satisfying x > 0) are called positive, and those which are to the left of 0 (thus those that satisfy x < 0) are negative. So 2∗ and ( 1 )∗ are negative, while all nonzero fractions are 3 positive, but if y is a negative number to begin with, y ∗ would be positive. The number 0 is, by definition, neither positive nor negative. As is well-known, a 1 1 number such as 2∗ is normally written as −2 and ( 3 )∗ as − 3 , and that the "−" sign in front of −2 is called the negative sign. However, it is better to avoid mentioning the negative sign until we get to subtraction, because we should develop one concept at a time. For teachers' need in the classroom, it would be a good idea to begin the discussion of the arithmetic of rational numbers with a concrete approach to the addition of rational numbers. To this end, define a vector to be a segment on the number line together with a designation of one of its two endpoints as a starting point and the other as an endpoint. We will continue to refer to the length of the segment as the length of the vector, and call the vector left-pointing if the endpoint is to the left of the starting point, right-pointing if the endpoint is to the right of the starting point. The direction of a vector refers to whether it is left-pointing or right-pointing. We denote vectors by placing an arrow above the letter, e.g., A, x, etc., and in pictures we put an arrowhead at the endpoint of a
47
vector to indicate its direction. For example, the vector K below is left-pointing and has length 1, with a starting point at 1∗ and an endpoint at 2∗ , while the vector L is right-pointing and has length 2, with a starting point at 0 and an endpoint at 2. 3∗ 2∗
'
1∗ K
0
1 L
2
E
3
By definition, two vectors being equal means exactly that they have the same starting point, the same length, and the same direction. For the purpose of discussing the addition of rational numbers, we can further simplify matters by restricting attention to a special class of vectors. Let x be a rational number, then we define the vector x to be one with starting point at 0 and endpoint at x. It follows from the definition that, if x is a nonzero fraction, then the segment of the vector x is exactly [0, x]. Here are two examples of vectors arising from rational numbers: 4∗ 3∗
'
2∗ → 3∗
1∗
0
1 → 1.5
1.5
2
E
With this notation understood, we now describe how to add such vectors. Given x and y, where x and y are two rational numbers, the sum vector x + y is, by definition, the vector whose starting point is 0, and whose endpoint is obtained as follows: slide y along the number line until its starting point (which is 0 ) is at x, then the endpoint of y in this new position is by definition the endpoint of x + y. For example, if x and y are rational numbers, as shown: 0
' E
y
48
x
Then, by definition, x + y is the point as indicated, 0
' E
x+y
x
We are now in a position to define the addition of rational numbers. The sum x + y for any two rational numbers x and y is by definition the endpoint of the vector x + y. In other words, x + y = the endpoint of x + y. Put another way, x + y is defined to be the point on the number line so that its −→ corresponding vector (x+y) satisfies: −→ (x+y)= x + y. We proceed to prove that the addition of rational numbers is commutative, i.e., x + y = y + x for all rational numbers. Of course this is equivalent to checking x + y = y + x. Remembering that two vectors are equal if and only if they have the same length and the same direction, we simply check that x + y and y + x do have the same length and same direction. The checking is straightforward. One can also prove that the addition of rational numbers is associative, i.e., (x + y) + z = x + (y + z) for all rational numbers x, y, z. However the reasoning this time is much more tedious and not so instructive. With this definition of the addition of rational numbers, one can prove in a hands-on manner the following basic facts for all positive fractions s and t:
Because s∗ + t = t + s∗ , by the commutative law of addition, the above four cases exhaust all the possibilities of the addition of any two rational numbers. We have just explicitly determined how to add any two rational numbers in terms of the addition and subtraction of fractions. We now must confront the fact that rational numbers are on a higher level of abstraction than fractions. A fact not mentioned in the brief discussion of fractions is that the addition and multiplication of fractions satisfy the associative, commutative, and distributive laws, but now things are going to change. We have just brought out the commutativity and associativity of addition among rational numbers. At this point, these laws must come to the forefront, because while the addition of rational numbers can be directly defined using the concept of vectors, there will be no such analog for multiplication. For the latter, we have to approach it from a different vantage point. Therefore, to prepare for multiplication, we forget the preceding definition of addition in terms of vectors and start from the beginning. We now take the attitude that although we do not know what the negative numbers are, the collection of rational numbers simply "must" satisfy the associative, commutative, and distributive laws with respect to addition and multiplication. Historically, this was what happened, and of course our intellectual inertia welcomes the status quo! Such being the case, one reasonable way to develop the addition of rational numbers is to make three fundamental assumptions about the addition of rational numbers at the outset. The first two fundamental assumptions are entirely noncontroversial: (A1) Given any two rational numbers x and y, there is a way to add these to get another rational number x + y so that, if x and y are fractions, x+y is the same as the usual sum of fractions. Furthermore, this addition of rational numbers satisfies the associative and commutative laws. (A2) x + 0 = x for any rational number x. The last assumption explicitly prescribes the role for all negative fractions: (A3) If x is any rational number, x + x∗ = 0.
50
On the basis of (A1)–(A3), one can proceed to compute the sum of two rational numbers in terms of the addition and subtraction of fractions as before. Let s and t be any two positive fractions. By (A1), s + t = the old addition of fractions. Then one can prove, with some effort, that (A1)–(A3) imply that
s∗ + t∗ = (s + t)∗ s + t∗ = (s − t) s + t∗ = (t − s)∗
if s ≥ t if s < t
We now pause to amplify on the second equality above by rewriting it as s − t = s + t∗ when s ≥ t.
The ordinary fraction subtraction s − t now becomes the addition of s and t∗ . This fact prompts us to define, in general, the subtraction between any two rational numbers x and y to mean: x − y = x + y∗ Note the obvious fact that the meaning of the subtraction of (say) the two rational 6 3 numbers 5 − 4 is, according to this definition, 6 3 + 5 4
∗
def
which, on account of "s + t∗ = (s − t) if s ≥ t", is just the fraction subtraction 6 3 5 − 4 . More generally, when x, y are fractions and x ≥ y, the meaning of x − y as a subtraction of rational numbers coincides, according to this definition, with the old meaning of subtracting fractions. In other words, we have not created a new concept of subtraction, merely made it more comprehensive. To repeat, 6 3 5 − 4 has exactly the same meaning whether we look at it as a subtraction between fractions or between rational numbers; this is reassuring. On the other hand, we are now free to do a subtraction between any two fractions such as 3 6 4 − 5 , whereas before (i.e., in item (B), Fractions) we could not carry out the subtraction because the first fraction is smaller than the second. We now see for the first time the advantage of having rational numbers available: we can as
51
freely subtract any two fractions as we add them. But this goes further, because we can even subtract any two rational numbers. The main message to come out of this definition is, however, the fact that subtraction is just a different way of writing addition among rational numbers. As a consequence of the definition of x − y, we have 0 − y = y∗ because 0 + y ∗ = y ∗ . Common sense dictates that we should abbreviate 0 − y to −y. So we have −y = y ∗ It is only at this point that we can abandon the notation of y ∗ and replace it by −y. Many of the preceding equalities will now assume a more familiar appearance, e.g., from x∗∗ = x for any rational number x, we get −(−x) = x, and from x∗ + y ∗ = (x + y)∗ , we get −(x + y) = −x − y
We now come to the multiplication of rational numbers, and we see the payoff from the more abstract approach to fraction addition. For multiplication, we make the following similar fundamental assumptions that (M1) Given any two rational numbers x and y, there is a way to multiply them to get another rational number xy so that, if x and y are fractions, xy is the usual product of fractions. Furthermore, this multiplication of rational numbers satisfies the associative, commutative, and distributive laws. (M2) If x is any rational number, then 1 · x = x. We note that (M2) must be an assumption because we would not know what 1× 5∗ means without (M2). The equally "obvious" fact, which is the multiplicative counterpart of (A2), to the effect that (M3) 0 · x = 0 for any x ∈ Q.
52
turns out to be provable once (M1) and (M2) are assumed to be true. We want to know explicitly how to multiply rational numbers. Thus let x, y be rational numbers. What is xy? If x = 0 or y = 0, we have just seen from (M3) that xy = 0. We may therefore assume both x and y to be nonzero, so that each is either a fraction, or the negative of a fraction. Letting s, t be positive fractions, one can prove on the basis of (M1)–(M3): (−s)t s(−t) (−s)(−t) = = = −(st) −(st) st
1 (e.g., (−3)( 2 ) = − 3 ) 2 3 (e.g., 3 (− 1 ) = − 2 ) 2
(e.g., (− 1 )(− 1 ) = 2 5
1 10 )
Since we already know how to multiply the fractions s and t, we have completely described the product of rational numbers. The last item, that if s and t are fractions then (−s)(−t) = st, is such a big part of school mathematics education that it is worthwhile to go over at least a special case of it. When students are puzzled by this phenomenon, the disbelief centers on how the product of two negatives can make a positive. The pressing need in this situation is most likely that of winning the psychological battle. So we propose to use a simple example to demonstrate why such a phenomenon is inevitable. Thus we will give the reason why (−1)(−1) = 1 Let us concentrate on the part of this assertion that says (−1)(−1) is a positive number. It would be nice if we can demonstrate this though a direct computation of the following type: we know (−3) − (−8) is positive because we can use the definition of subtraction and the above rules for adding rational numbers to conclude that (−3) − (−8) = (−3) + (−8)∗ = 3∗ + (8∗ )∗ = 3∗ + 8 = 8 + 3∗ = (8 − 3) = 5 This is a most satisfying proof because we see explicitly how the answer "5" comes out of a direct computation. The proof leaves no room for doubt. However, this kind of proof is not always around, and we are sometimes forced to use an indirect method to find the answer. To give this line of thinking some context, you may remember what you learned in your school chemistry: if you have to find out
53
whether a bottle of liquid is acidic or alkaline, the best scenario would be that there is clear label on the bottle stating it is HCl or ammonia. If not, then you would have to resort to an indirect method by dipping a blue litmus strip in the liquid: if the strip turns red, then it is an acid. So you have to trust the litmus paper and allow it to give you the needed information indirectly. It is the same with (−1)(−1). There is no known explicit computation with (−1)(−1) so that a positive number pops out at the end of the computation, but a possible "litmus test" in this case is to add to (−1)(−1) a negative number. If the answer is either 0 or positive, then you'd have to agree that (−1)(−1) is a positive number. This is exactly what we are going to do. So we are going to test the positivity of (−1)(−1) by adding to it the negative number −1. Why −1 and not some other negative number? This comes from experience and some common sense; one way is to ask yourself why not −1 ? After all, it is natural to think of −1 in this particular context. In any case, we are going to apply the distributive law (assumption (M1)) to this sum and get: (−1)(−1) + (−1) = (−1)(−1) + 1 · (−1) = {(−1) + 1}(−1) = 0 · (−1) = 0, where the last equality is by (M3). This therefore shows that, if we believe in the distributive law for rational numbers, it must be that (−1)(−1) is positive. In fact, we know a bit more: if (−1)(−1) added to −1 is 0, then (−1)(−1) has to be 1. In other words, (−1)(−1) = 1. The fact that the distributive law holds for rational numbers is also responsible for the general assertion that (−s)(−t) = st. The concept of the division of rational numbers is the same as that of dividing whole numbers or dividing fractions. For two rational numbers x and y, with x y = 0, y is by definition the rational number z so that x = zy. As in the case of fractions, the existence and uniqueness of such a z must be proved. Assuming this, we can now clear up a standard confusion in the study of rational numbers mentioned above, namely, the reason why the following equalities are true: 3 −7 = −3 7 3 =− . 7
3 −7
First let C = − 3 . We want to prove that 7
= C. This would be true, by
54
definition, if we can prove 3 = C × (−7), and this is so because 3 3 C × (−7) = (− ) × (−7) = ( )(7) = 3 7 7 where we have made use of (−a)(−b) = ab for all fractions. Of course this proves 3 3 −3 3 −7 = − 7 . In a similar manner, we can prove 7 = − 7 . More generally, the same reasoning supports the assertion that if k and whole numbers and = 0, then −k = k − = − k and −k k = . − are
We may also summarize these two formulas in the following statement: for any two integers a and b, with b = 0, −a a a = =− . b −b b This formula is well-nigh indispensable in everyday computations with rational numbers. In particular, it implies that every rational number can be written as the quotient of two integers. Thus, the rational number −
9 7
is equal to
−9 7
or
9 −7 .
The concept of complex fractions has a counterpart in rational numbers, of course. For lack of a better name, we call them rational quotients, and as in the case of complex fractions, rational quotients can be added, subtracted, multiplied, and divided as if their numerators and denominators were whole numbers. Finally, to compare rational numbers, recall the definition of x < y between two rational numbers x and y: it means x is to the left of y on the number line. x y
The following inequalities are basic to any discussion of rational numbers and therefore belong to middle school mathematics. Here, x, y, z are rational numbers, and the symbol " ⇐⇒" stands for "is equivalent to":
Then the relative positions of 2x and 2y do not change, but each is pushed further to the right of 0: 0 2x 2y
If we reflect this picture across 0, we get the following: −2y −2x 0 2x 2y
We see that −2y is now to the left of −2x, so that −2y < −2x, as claimed. Obviously, this consideration is essentially unchanged if the number 2 is replaced by any negative number z. (D) Geometry Our teachers are generally ill-prepared on the subject of geometry (cf. [IMAGES]). They are often misled into believing that introductory geometry is nothing more than one big vocabulary test, and not a very precise
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vocabulary at that. We have to impress on them, first of all, that there is a need for this vocabulary to be precise, and secondly, that geometry is about the reasoning underlying the precise study of spatial figures rather than just the vocabulary. Precision in the vocabulary is necessary because it is only through this vocabulary that we can transcribe intuitive spatial information into precise mathematics, and it is entirely on this vocabulary that we base our reasoning. The definitions of common concepts such as "angle", "convex figures", "polygons", etc., are anything but obvious. We also have to impress on them the fact that we draw a distinction here between what they as teachers should know and what they teach their students in K-8, especially in grades 4-5. The precision that they as teachers should learn may not literally translate into suitable classroom material in upper elementary and middle school as it might overwhelm students at this stage of their mathematical development. But it behooves teachers to learn such precision, because they must know the whole truth before they can judiciously hide unpleasant details in the service of good teaching. Part of the professional development in geometry would ideally include lots of drawings-byhand and some hands-on activities such as the construction of regular polyhedra. There has to be an integration of the direct spatial input with the verbal-analytic output in a geometric lesson. An important component of K–8 geometry (one that has not yet been fully implemented in the classroom) is familiarity with the three basic rigid motions in the plane: rotation, translation, and reflection (teachers should be steered away from the uncivilized terminology of "turn, glide, and flip"). Certainly, professional development must give precise definitions of these concepts, but in this case, the professional development may include the information that, with the availability of (overhead projector) transparencies, these concepts can be graphically demonstrated so that, in the school classroom, the precise definitions may be soft-pedaled in exchange for a tactile and intuitive understanding. The said demonstration consists of making identical drawings on two pieces of transparencies, preferably in different colors. By moving one against the other, the effects of basic rigid motions, and compositions thereof, can be tellingly displayed, and students get a sound conception of what these rigid motions do. It is in this context that we recommend that in a middle school classroom, the quite sophisticated precise definitions of basic rigid motion be soft-pedaled. The hope is that students will reprise these concepts in a high school geometry course, so
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that if they attain at least a good intuitive knowledge in the middle grades, they will gain a better understanding the second time around. In terms of these basic rigid motions, congruence can now be defined as a finite composition of such. (See Example 4 of Part I.) Teachers should learn that, while "same size, same shape" is a good sales pitch about congruence for the general public, it should not be offered as a mathematical definition because it does not conform to the basic characteristic of precision. There is an urgent need in school mathematics to replace "same size, same shape" with the above definition of congruence that is correct and is also something that students can directly experiment with. The next important topic is dilation. First of all, teachers should be convinced of the feasibility of teaching this concept in middle school. For example, it should be pointed out that a precise definition of dilation would allow students to magnify any picture by any scale factor, if enough sampling points of the original picture are chosen. (See Example 5 of Part I.) Such magnification (or contraction) activities have never ceased to impress students. Therefore teaching the correct definition of a dilation would not be a hard sell. Professional development can build on this fact. Of course the reason one needs dilation is that the concept of similarity can now be correctly defined as the composition of a dilation and a congruence. The error in school textbooks, defining similarity as "same shape but not necessarily same size", must be corrected as soon as possible. The reason for the critical need of a definition of similarity is that a working knowledge of similar triangles is absolutely essential for students to achieve algebra. Without this knowledge, they would have no hope of understanding the interplay between a linear equation of two variables and its graph, which is a major topic in beginning algebra. This means our teachers of middle school must be completely at ease with similar triangles and know how to exploit the same, and they must know the underlying reasoning. The professional development on this topic may include the information that, while teachers should know the reasoning (i.e., proofs) behind the AAA (three pairs of equal angles) and SAS (two pairs of proportional sides and a pair of equal included angles) criteria for similarity, students in middle school can get by with less. Students can afford to learn to use these theoretical tools first and wait for their explanations later.
58
This is standard practice in mathematics education (e.g., the teaching of calculus without epsilon-delta). Professional development should assure teachers of this fact so that they do not feel overwhelmed by the need to teach all the proofs about similarity, something that even our high school teachers may find difficult. The goal is to equip middle school teachers with this knowledge so that they can better instruct their students about similarity. Another important topic in the teaching of elementary geometry is the concept of measurement, which leads to the standard mensuration formulas about area of triangles, circumference of circles, etc. Conceptually, there is no difference between length, area, or volume. If we let "measurement" stand for any of these three concepts, then on the basis of the following three entirely reasonable statements, all the standard mensuration formulas can be proved: (1) Measurement is the same for congruent sets. (2) Measurement is additive, in the sense that if two sets are disjoint except at their respective boundaries, then the measurement of the union is the sum of the measurements of the two sets. (3) If a set S is the limit of a sequence {Si } in an appropriate sense, then the measurement of S is the limit of the measurements of the {Si }. Of course (3) has to be carefully qualified, and any discussion of "limit" has to remain intuitive, but if experience is any guide, such an approach to limit at the middle school level does not seem to be a hindrance to learning. As is wellknown, the introduction of limit at this juncture is necessary if the circumference and area of a circle are to be meaningfully computed. The noteworthy feature of these three assumptions lies not in (3), but, rather, in how congruence enters the discussion of measurement through (1). The fact that the concept of congruence underlies the concept of measurement has not been sufficiently emphasized in school mathematics, but it should be. This is another example of the coherence of mathematics. Incidentally, the important role played by congruence in the study of measurements is one reason why congruence must be correctly defined. It should be mentioned that while the number π can be defined in many ways, a strong recommendation for school mathematics is to define it, not as circumference divided by the diameter, but as the area of the unit circle. The former does not lend itself to any hands-on experiments to determine its value
59
with any precision, whereas the latter does. Using accurate grid papers (with small grids), one can approximate the area of a circle by counting the number of grids in it together with elementary estimation of those only partially in it, and the value of π estimated by this method usually comes out to be within 0.05 of the exact value. (E) Algebra It can be argued that the most basic aspect of the learning of algebra is the fluent use of symbols. Unfortunately, if textbooks are any guide, students' attempt to learn about symbols is at present hindered by the need to master the concept of a variable. There are two reasons why the concept of a variable unnecessarily obstructs learning. The first one is that the mathematics education literature, including textbooks, does not make explicit what a "variable" is. It is sometimes described as a quantity that changes or varies. At other times, it is asserted that students' understanding of this concept should be beyond recognizing that letters can be used to stand for unknown numbers in equations, but it does not say what exactly lies "beyond" this recognition. A second reason is that while mathematicians use the terminology of a "variable" informally and often, there is no mathematical concept called a "variable". The closest that comes mind is "an element in the domain of definition of a function", or "the indeterminate of the polynomial ring R[x]", but certainly nothing varies in mathematics. The first task in the professional development of algebra therefore has to be to disabuse prospective teachers of this notion of a "variable" that they acquired in their K–12 education. There is absolutely no need for it. For further discussions of how to handle the pedagogical issue of the use of symbols and what a "variable" is, see pp. 3–6 of [Schmid-Wu]. In summary, the important thing at the beginning algebra is to get used to using symbols to represent numbers and to compute with them. It is not necessary to worry about what a "variable" means. However, it should be also pointed out that, the language of "variable" being entrenched in mathematics as it is, it would be to our advantage to follow the common usage and use it informally when it is convenient. But each time we do, we will be explicit about what the word stands (though it will not be anything that "varies"). Let a letter x stand for a number, in the same way that the pronoun "he" stands for a man. Then any (algebraic) expression in x is a number, and all the knowledge accumulated about rational numbers can be brought to bear on
60
such expressions. There is a caveat, however. Because all we know about such an x is a number without any knowledge of its exact value, computations with expressions in x must then be done using only all the rules we know to be true for all numbers, namely, the associative and commutative laws and the distributive law. Doing computations not with specific numbers but with an arbitrary number brings into focus the concept of generality. For this reason, beginning algebra is generalized arithmetic. Nevertheless, arithmetic it is, and despite students' initial unfamiliarity with the presence of a large number of symbols, they will soon get used to computing with polynomials or rational expressions as ordinary numbers. Note that students who are uncomfortable with ordinary number computations to begin with may be made even more uncomfortable at this juncture.This underscores the importance of a firm grasp of rational numbers for the learning of algebra. (Cf. [Wu] 2001.) For example, the following addition of rational expressions in a number x can be carried out as with rational quotients, x2 6 + 2 (3x4 + x + 2) (x + 5) = x2 (x2 + 5) + 6(3x4 + x + 2) , (3x4 + x + 2)(x2 + 5)
because if k, , m, n stand, respectively, for the numbers x2 , (3x4 + x + 2), 6, and (x2 + 5), then each of these is a rational number and the equality becomes nothing more that the usual formula for the addition of rational quotients: k + m kn + m = n n
(See the comments on rational quotients in item (C) on Rational numbers above.) Three remarks about the preceding paragraph should be made in the context of professional development. The first is that consideration of the arithmetic of rational expressions confirms why the arithmetic of complex fractions and rational quotients are indispensable to the learning of algebra. Teachers need to be aware of this fact for their own teaching of fractions and rational numbers. A second one is the reference above to x as a number. In school mathematics, the only kind of numbers treated with any thoroughness are the rational numbers. Irrational numbers are basically no more than a name. Unfortunately, it is not in the tradition of school mathematics to be explicit about the restriction to only rational numbers in mathematical discussions about real numbers. For example,
61
the preceding paragraph implicitly assumes that even if x is an irrational number, the addition of the two rational expressions above will continue to hold. This is indeed correct on account of advanced considerations about the "extension of continuous functions from rational numbers to real numbers". The explanation of the phrase in quotes is beyond the level of normal professional development for middle school teachers, but we are nevertheless obligated to make teachers aware of this extrapolation from rational numbers to all numbers. One can succinctly formulate this extrapolation as FASM, the Fundamental Assumption of School Mathematics (see [Wu2002b]): All the information about the arithmetic operations on fractions can be extrapolated to all real numbers. A third and final remark is that if we let x be a whole number, then the above addition x2 6 + 2 (3x4 + x + 2) x + 5 becomes an addition of two (ordinary) fractions because the numerators and denominators are whole numbers. Notice therefore how the addition was carried out, which is to use the basic formula k + m kn + m = n n
without worrying about the LCM of the whole numbers 3x4 + x + 2 and x2 + 5. If elementary school teachers can take note of this fact in algebra, then they will realize how misguided it really is to teach the addition of fractions using the LCM of the denominators, which is how most school textbooks still teach it. If we do not want to mislead students with this kind of defective information, it would be most helpful if teachers can see through the defect. Incidentally, this is one reason why we want teachers to know the mathematics of several grades beyond what they teach (cf. [NMPa], Recommendation 19 on page xxi.) It should be pointed out that, if the letters x and y are just numbers, then the distributive law gives xn+1 − y n+1 = (x − y)(xn + xn−1 y + xn−2 y 2 + xn−3 y 3 + · · · + xy n−1 + y n ) for any two numbers x and y, and any positive integer n.
62
Because this equality of these two expressions in x and y is valid for all numbers x and y, we call the equality an identity. Letting y = 1, we get another identity: xn+1 − 1 = (x − 1)(xn + xn−1 + xn−2 + · · · + x2 + x + 1) for all numbers x and for all positive integers n. If x = 1, multiplying both sides 1 by the number x−1 and switching the left and the right sides give: 1 + x + x2 + x3 + · · · + xn = xn+1 − 1 x−1
for any number x = 1, and for any positive integer n. This is of course the so-called summation of the finite geometric series. In a school classroom, one might teach this summation formula by first doing a few concrete cases such as n = 2, n = 3 and n = 4, before doing it for a general n. This summation formula is usually taken up near the end of the study of algebra in high school. We have seen that there is no reason for the delay, all the more so because this formula is important in so many areas of mathematics. A major topic in beginning algebra is the relationship between a linear equation in two variables ax + by = c and its graph. To the extent that our teachers learned from their K–12 school textbooks, there would be many gaps in teachers' knowledge about these equations. The first is a correct definition of the slope of a line L. It needs to be shown that the slope of L defined by two chosen points P and Q on L is in fact independent of the choice of P and Q. In this case, it is not merely correctness for its own sake. Knowing this independence leads to the awareness that, in each situation, one can choose the two points most suitable to one's purpose for the computation of the slope. Sometimes, being able to make such a choice is the difference between success and failure. But this independence cannot be proved without knowing the AAA criterion for the similarity of triangles (i.e., two triangles with three pairs of equal angles are similar), and this is the reason similarity must be taught correctly before taking up algebra. A second gap is the precise definition of the graph of ax + by = c as the set of all the points (x , y ) whose coordinates satisfy the equation, i.e., ax + by = c. Without explicitly invoking this definition, it would be impossible to prove the basic theorem of linear equations in two variables, to the effect that the graph of ax + by = c is a line, and any line is the graph of some linear equation of two
63
variables. The lack of emphasis in enunciating the definition of the graph of an equation goes hand-in-hand with the absence of this proof in most algebra textbooks. Such a proof depends strongly on knowing the precise definition of the graph of an equation and on knowing when two triangles are similar. Students who understand the details of this proof will have a good grasp of the genesis of the many forms of the equation of a line (point-slope form, slope-intercept form, etc.) that satisfies some prescribed conditions, e.g., passing through two given points; they will have no need to memorize these different forms by brute force. At the moment, anecdotal evidence suggests that the relationship between a linear equation and its graph remains a black box to many teachers and students; if this can be verified by research, it would mark a significant progress in the teaching and learning of algebra. Associated with linear equations in two variables are linear inequalities. Again, one must first give a precise definition of the graph of an inequality and then prove that such a graph is a half-plane. There are many ways to handle this theorem. A drastic way to cut through the subtleties is to simply define a halfplane to be the graph of a linear inequality and then give many examples and ample discussions to make this drastic step reasonable. A more reasonable alternative is to define a half-plane of a line L which is not vertical to be all the points above L or all the points below L, and then prove the theorem. (The definition of the half-planes of a vertical line is trivial.) However, no matter which approach is adopted, it is not an acceptable way to teach mathematics by not defining either the graph of an inequality or a half-plane, and then engaging in the wishful thinking that, by drawing a few pictures, the reader would automatically understand how to define these terms and also buy into the theorem. The uphill battle one must fight in the professional development of algebra therefore includes the need to convince teachers not to engage in such practices and to learn a new way to deal with this topic. Many algebra curricula now take up linear programming as an application of linear inequalities. The fact that a linear function assumes it maximum and minimum in a convex polygonal region at a vertex then requires a careful explanation. For this purpose, it is all the more reason to have a precise definition of the graph of a linear inequality and that of a half-plane.
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The subject of simultaneous linear equations (to be called linear systems) is a straightforward study of the interplay between two linear equations, or what is the same thing, the interplay between two lines, but its simplicity is often compromised in school texts. There, the meaning of the solution of a linear system is almost never explicitly given, with the result that it is not used to explain why the point of intersection of the lines defined by the individual equations provides a solution of the linear system then cannot be given. Students are told to use graphing calculators to get the solution of a linear system, but they are not told why the graphing calculator gives the right answer. This is a new paradigm of learning-by-rote, and is one that we should strive to eliminate from the school classrooms. To this end, our teachers must receive careful instruction on the definitions in question as well as the associated explanations. There is another misconception associated with linear systems. The usual method of solution of a linear system by substitution, due to misinformation from textbooks, has been interpreted as an exercise in the symbolic manipulation of variables. When done correctly, however, the method of substitution is strictly a computation with numbers, no more and no less, and there is no need in this context to worry about the purported complex meaning of the equal sign when applied to variables. Indeed, what the substitution method does is not to produce a solution, but rather, starting with the assumption that a solution (x0 , y0 ) exists, it shows what the values of x0 and y0 must be. Then one substitutes these values into the original linear system to verify easily that they are solutions. Explications of such subtleties have to be a necessary component of any reasonable professional development in algebra. The next major topic in algebra is the concept of a function, its definition and the study of linear and quadratic functions. Again, one should insist on explicit definition of the graph of a function; for real-valued function f of one variable, its graph is the subset of the plane consisting of all ordered pairs {(x, f (x))} where x is a member of the domain of f . The graphs of linear functions, f (x) = cx + k, are lines, and this follows from the work on graphs of linear equations in two variables because the graph of the function f is seen to be the graph of the linear equation in two variables y = cx + k. A special class of linear functions, those without constant term k, are especially important in middle school mathematics. They underlie all considerations of constant rate. Constant speed,
65
for example, is the statement that there is a constant v, so that if the distance traveled from time 0 to time t is f (t), then f (t) = vt. It also underlies all the problems connected with proportional reasoning. This then calls for a little soul-searching in this connection. At the moment, there appears to be some misconception about the formulation of mathematical problems. Consider a prototypical proportional-reasoning problem such as: A group of 8 people are going camping for three days and need to carry their own water. They read in a guide book that 12.5 liters are needed for a party of 5 persons for 1 day. How much water should they carry? It should be clear that this problem cannot be done without first assuming that everybody drinks the same amount of water each day. There is nothing obvious about this assumption, because even young kids can see that some people drink lots of water and others very little. If we believe that mathematics is precise, then precision demands that this assumption be made explicit. An alternative to the question, "How much water should they carry?", is to ask how to make a rough estimate of the amount of water they should carry if we simplify matters by assuming that everybody drinks the same amount everyday. Once we have this assumption, let f be the function defined on the whole numbers so that f (n) is the amount of water n people drink each day. Then the assumption gives f (n) = cn for some constant c, where c is the amount of water each person is assumed to drink per day. A common practice is to now allow the symbol n to stand for any number, and not just a whole number, so that f (n) becomes a linear function without constant term. With the given data that f (5) = 12.5, we want the number 3f (8). From the former, we get c = 2.5, so the answer is 3 × 2.5 × 8 = 60 liters. The main point is, however, that if proportional reasoning is about "understanding the underlying relationships in a proportional situation and working with these relationships" ([NRC 2001], p. 241), then the proportional relationship must be made explicit for students as otherwise students would be groping in the dark without a clue. Professional development should make this point very clear: guesswork is not to be confused with conceptual understanding, and there is no ground for assuming that every student "understands" that all people drink the same amount of water everyday. Once these ground rules are understood, a discussion of word problems related to proportional reasoning from the point of view of linear functions should be
66
both revealing and rewarding. From linear functions we go to quadratic ones. The graph of a linear function is a line, but what is the graph of a quadratic function? If by some good fortune we know that the quadratic function is presented to us in the form of f (x) = a(x + p)2 + q, where a and q are fixed numbers, then one can picture the graph of f without too much effort, as follows. From f (−p + s) = f (−p − s), we see that for all numbers s, the point (−p−s, f (−p−s)) and the point (−p+s, f (−p+s)) are symmetric relative to the vertical line x = −p. Therefore the graph of f has an axis of reflection symmetry along the vertical line x = −p, and the graph has its lowest (resp., highest) point at (−p, f (−p)), if a > 0 (resp., a < 0). Of course, f (−p) = q. So at least for simple quadratic functions expressible as f (x) = a(x + p)2 + q, the graph is completely understood, and therewith, the function itself is completely understood. In fact, we can trivially read off from the equality f (x) = a(x + p)2 + q where the function is equal to 0, namely, −p ± − q a
The fundamental theorem about quadratic functions is that, by the technique of completing the square, every quadratic function can be written in the form f (x) = a(x + p)2 + q for fixed numbers a, p, and q. Included in this statement is the quadratic formula for the roots (zeros) of f , but the whole discussion make it perfectly clear that the technique of completing the square is the key to the understanding of quadratic functions. One can then go on to discuss the relationship between roots and the factoring of a quadratic function, and also the relationship between the roots and the coefficients a, b, c in f (x) = ax2 + bx + c. In particular, teachers should be aware that the quadratic formula trivializes, at least in principle, the problem of factoring quadratic polynomials, because one proves (the far from obvious statement) that if r1 , r2 are the roots of a quadratic polynomial f , then f (x) = a(x − r1 )(x − r2 ) for some fixed number a, and this is a factoring of f . Since the quadratic formula provides the values of the roots, the factoring immediately follows. The implication of this discussion for professional development is therefore quite clear: our algebra teachers have to understand this aspect of the quadratic formula in order for them to teach the factoring of trinomials with the proper perspective. The availability of quadratic functions enlarges the range of word problems.
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Their discussion should be an integral part of professional development. If we look ahead into the high school curriculum, we see that introductory algebra is inextricably tied to the materials in high school mathematics as a whole. We have already emphasized the important role of similar triangles in the discussion of linear equations of two variables. For teachers to be comfortable teaching about such linear equations, they would have to learn the proofs of the basic criteria for similar triangles. As is well-known, similarity is the deepest part of plane geometry. Thus a teacher who wants to teach introductory algebra well should be at least familiar with high school geometry. Furthermore, we have seen that linear and quadratic functions are a staple of introductory algebra. Teachers cannot, however, teach about these functions if this is all they know. They need a reservoir of knowledge about a few other standard functions as well, e.g., higher degree polynomial functions, exponential functions, logarithmic functions, periodic functions. For polynomial functions in general, teachers need to be familiar with the most basic facts, such as the Fundamental Theorem of Algebra and its implications on the factorization of polynomials, especially the factorization of real polynomials. But these already comprise all the main topics in the school algebra curriculum. Finally, among the most profound applications of similarity is the definition of the trigonometric functions, and these are the basic building blocks of periodic functions. A teacher of beginning algebra should therefore also know something about trigonometry. If we can draw any conclusion from the preceding discussion, it must be that a middle school teacher who teaches beginning algebra should have at least a good mastery of the high school mathematics curriculum. (F) Probability and statistics It has long been recognized by some mathematicians and statisticians that K–6 is not the place to teach serious statistics. The recent publication of Curriculum Focal Points by the National Council of Teachers of Mathematics ([NCTM] 2006) has finally acknowledged this fact. There is no harm in discussing the basic notions such as mode, mean, and median to liven up the discussion of number facts from time to time, but teachers should be made aware that meaningful statistics appears only in high school or beyond, when there is sufficient mathematical preparation to support such a discussion. They should also be alerted to the senseless practice in standardized
68
tests of asking for the mode of a small number of items. The concept of mode is meaningful only when a large number of items is involved. The basic concepts of probability can and should be discussed in middle school, of course, such as the fact that probabilities are numbers between 0 and 1, the concept of a sample space, the relation between theoretical probability and the relative frequency of an event. The teaching of simple facts concerning binomial coefficients and elementary combinations and permutations would also enrich the curriculum of middle school. | 677.169 | 1 |
Algebra 2H
After each quiz, you will be writing a journal entry. They need to be written in complete
sentences and you need to use proper grammar. Make sure you are using your own
words! Please make sure you proof read it and turn in your best product pos
THE PLAN
We are going to flip our classroom for the next chapter. That means you
will watch lectures at home off of youtube and then work on hw problems and
a collaborative project during class time. This is a work in progress suggestions are welcome as w | 677.169 | 1 |
Cpm homework help cc3
When Working on Problems from the textbook, please refer to the CPM Homework Help link below with hints to work through difficult problems.
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Microsoft Word - CC3 Selected Answers ch 7.docx Author: Carol Cho.
Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus.CPM is a non-profit corporation dedicated to teaching more students more mathematics by providing.
Accessing the CPM Online Textbook Your child will be given the following URL to sign up for the CC3 e-book.Cpm homework help cc3, outline of essay for middle school, writing essays poetry - time to study.Homeworkhelp Cpm The reasons that make students search for cpm homework help are many and varied. | 677.169 | 1 |
will cover the mathematical theory and analysis of simple games without chance moves. This course explores the mathematical theory of two player games without chance moves. We will cover simplifying games, determining when games are equivalent to numbers, and impartial games.This is a course about the Fibonacci numbers, the golden ratio, and their intimate relationship. In this course, we learn the origin of the Fibonacci numbers and the golden ratio, and derive a formula to compute any Fibonacci number from powers of the golden ratio.
This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus.
This course covers mathematical topics in trigonometry. Trigonometry is the study of triangle angles and lengths, but trigonometric functions have far reaching applications beyond simple studies of triangles. This course is designed to help prepare students to enroll for a first semester course in single variable calculusThis course introduces simple and multiple linear regression models. These models allow you to assess the relationship between variables in a data set and a continuous response variable. Is there a relationship between the physical attractiveness of a professor and their student evaluation scores? Can we predict the test score for a child based on certain characteristics of his or her mother? In this course, you will learn the fundamental theory behind linear regression and, through data examples, learn to fit, examine, and utilize regression models to examine relationships between multiple variables, using the free statistical software R and RStudioLearn how to apply selected mathematical modelling methods to analyse big data in this free online course. Mathematics is everywhere, and with the rise of big data it becomes a useful tool when extracting information and analysing large datasets | 677.169 | 1 |
Synopses & Reviews
Synopsis
Convex Analysis is an emerging calculus of inequalities while Convex Optimization is its application. Analysis is the domain of the mathematician while Optimization belongs to the engineer. In layman's terms, the mathematical science of Optimization is a study of how to make a good choice when faced with conflicting requirements. The qualifier Convex means: when an optimal solution is found, then it is guaranteed to be a best solution; there is no better choice. As any convex optimization problem has geometric interpretation, this book is about convex geometry (with particular attention to distance geometry) and nonconvex, combinatorial, and geometrical problems that can be relaxed or transformed into convexity. A virtual flood of new applications follows by epiphany that many problems, presumed nonconvex, can be so transformed. This is a BLACK & WHITE paperback. A hardcover with full color interior, as originally conceived, is available at lulu.com/spotlight/dattorro | 677.169 | 1 |
Course 18C Mathematics with Computer Science
Mathematics and computer science are closely related fields. Problems in computer
science are often formalized and solved with mathematical methods. It is likely that
many important problems currently facing computer scientists will be solved by
researchers skilled in algebra, analysis, combinatorics, logic and/or probability
theory, as well as computer science.
The purpose of this program is to allow students to study a combination of these
mathematical areas and potential application areas in computer science. Required subjects
include linear algebra (18.06, 18.700 or 18.701) because it is so broadly used, and
discrete mathematics (18.062J or 18.200) to give experience with proofs and the
necessary tools for analyzing algorithms. The required subjects covering complexity
(18.404J or 18.400J) and algorithms (18.410J) provide an introduction to the most
theoretical aspects of computer science. We also require exposure to other areas
of computer science (6.031, 6.033, 6.034, or 6.036) where mathematical issues may
arise.
Some flexibility is allowed in this program. In particular, students may substitute
the more advanced subject 18.701 Algebra I for 18.06, and if they already have strong
theorem-proving skills, may substitute 18.211 for 18.062 or 18.200.
Required Subjects
18.03 or 18.034 (Differential Equations)
[sufficiently advanced students may substitute 18.152 or 18.303]
18.06, 18.700 or 18.701 (Linear Algebra)
18.410J (Design and Analysis of Algorithms)
6.0001 (Introduction to Computer Science and Programming)
6.006 (Introduction to Algorithms)
6.009 (Fundamentals of Programming)
[students who entered M.I.T. before September 2016 may use 6.01 for this requirement.]
Restricted Electives
Four additional 12-unit subjects from Course 18 and one additional subject of
at least 12 units from Course 6. The Course 6 subject may not be 6.042J or 6.00,
and may be a 6.S* subject only with permission of the Mathematics Department.
The overall program must consist of
subjects of essentially different
content and must include at least five Course 18 subjects with first decimal
digit one or higher.
Note: The program described above fully affects students entering MIT on or after
September 2016. Because changes in course 6 are being phased in over time,
current students may need some flexibility in transitioning from the earlier
18C requirements. Please consult the Department for any questions. | 677.169 | 1 |
In your own words, describe the special cases of integer programming and binary programming: what makes thes
different? Give an example of each, pointing out why they must be an integer or binary programming problem as
standard linear programming problem.
Spreadsheet Management Sience Advice
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I would recommend this course because the professor is nice and but the course is a little difficult, especially if you don't read the material. There is a lot of calculations involved and a lot of analyzing of spreadsheets and numbers.
Course highlights:
I have the gained the knowledge of using raw data to analyze results using spreadsheets and different functions. | 677.169 | 1 |
College Algebra: Exponential Growth and Decay
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College Algebra: Exponential Growth and Decay
One of the most common applications of logs and exponentials is using e (2.718) to calculate rates of growth or rates of decay. In this lesson, we will go through the model for exponential growth (e.g. compounding interest, population growth, etc) and the model for exponential decay (e.g. half-life problems for radioactive decay or medicinal effectiveness declines). In evaluating many of these problems, you'll use the identity e^ln A = A because the log function and the ln function are inverse functions. Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.
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Get free advice from education experts and Noodle community members. | 677.169 | 1 |
not only taxes a student's memory unduly but in variably leads to mechanical modes of study.
All
practical
teachers
know how few
students understand and
appreciate the more difficult parts of the theory. etc. Typical in this respect is the
treatment of factoring in
many
text-books
In this book
all
methods which are of
and which are applied in advanced work are given. and conse-
. Until recently the tendency was to multiply as far as possible.
giving to the student complete familiarity
with
all
the essentials of the subject. The entire study of algebra becomes a mechanical application of memorized
rules.
owing
has certain distinctive features.
however. are omitted. but "cases" that are taught only on account of tradition."
this book..
All unnecessary methods
and "cases" are
omitted.
and ingenuity
while the cultivation of the student's reasoning power is neglected.
"
While
in
many
respects
similar to the author's
to its peculiar aim.PREFACE
IN
this
book the attempt
while
still
is
made
to shorten the usual course
in algebra.
manufactured for this purpose. chief
:
among
These
which are the following
1.
specially
2.
Elementary Algebra. All parts of the theory whicJi are beyond the comprehension
of
the student or wliicli are logically
unsound are
omitted. in order to make every example a
social
case of a memorized method.
omissions serve not only practical but distinctly pedagogic " cases " ends.
Such a large number of methods. short-cuts that solve only examples
real value.
hence either book
4. as quadratic equations and
graphs. Moreover.
In regard
to
some other features of the book.
differ
With very few
from those
exceptions
all
the exer
cises in this
book
in the
"Elementary Alge-
bra".vi
PREFACE
quently hardly ever emphasize the theoretical aspect of alge bra.
may
be used to supplement the other. " The book is designed to meet the requirements for admis-
sion to our best universities
and
colleges. are placed early in the course. all proofs for the sign age
of the product of
of the binomial
3.
The best way to introduce a beginner to a new topic is to offer Lim a large number of simple exercises. especially problems and factoring. etc. in particular the
requirements of the College Entrance Examination Board.g. the following
may
be quoted from the author's "Elementary Algebra":
which
"Particular care has been bestowed upon those chapters in the customary courses offer the greatest difficulties to
the beginner. a great deal of the theory offered in the avertext-book is logically unsound . e. For the more ambitious student. there has been placed at the end of the book a collection of exercises which contains an abundance
of
more
difficult
work.
TJie exercises are slightly simpler than in the larger look. all elementary proofs theorem for fractional exponents. however. The presenwill be found to be tation of problems as given in Chapter
V
quite a departure from the customary way of treating the subject.
two negative numbers. This made it necessary to introduce the theory of proportions
.
Topics of practical importance. and it is hoped that this treatment will materially diminish the difficulty of this topic for young students.
enable students
who can devote only a minimum
This arrangement will of time to
algebra to study those subjects which are of such importance for further work.
McKinley
than one that gives him the number of Henry's marbles.
nobody would find the length Etna by such a method.
is
based principally upon the alge-
.
and they usually involve difficult numerical calculations."
Applications taken from geometry.
viz.PREFACE
vii
and graphical methods into the first year's work. and commercial are numerous.'
This topic has been preit is
sented in a simple. but the true study of algebra has not been sacrificed in order to make an impressive display of sham
life
applications. physics. " Graphical methods have not only a great practical value. The entire work in graphical methods has been so arranged that teachers who wish
a shorter course
may omit
these chapters. in
"
geometry
. based upon statistical abstracts. are
frequently arranged in sets that are algebraically uniform. the student will be able to utilize this knowledge where it is most
needed.
but they unquestionably furnish a very good antidote against 'the tendency of school algebra to degenerate into a mechanical application of
memorized
rules. an innovation which seems to mark a distinct gain from the pedagogical point
of view.
to solve a
It is
undoubtedly more interesting for a student
problem that results in the height of Mt. and
of the
hoped that some
modes of representation given
will be considered im-
provements upon the prevailing methods. Moreover.
of the Mississippi or the height of Mt. But on the other hand very few of such applied examples are
genuine applications of algebra. elementary way.
By studying proportions during the first year's work.
while in the usual course proportions
are studied a long time after their principal application. and hence the student is more easily led to do the work by rote
than when the arrangement
braic aspect of the problem. such examples.
ARTHUR SCHULTZE. 1910. desires to acknowledge his indebtedness to Mr.
NEW
YORK.
pupil's knowlso small that an extensive use of
The average
Hence the
field of
suitable for secondary school
tations.
William P.viii
PREFACE
problems relating to physics often
offer
It is true that
a field
for genuine applications of algebra. is such problems involves as a rule the teaching of physics by the
teacher of algebra.
April.
edge of physics.
. Manguse for the careful reading of the proofs and
many
valuable suggestions. however.
genuine applications of elementary algebra work seems to have certain limi-
but within these limits the author has attempted to
give as
many
The author
for
simple applied examples as possible.
= 3. 6.c) (a . 25.
24.6 -f c) (6
a
+ c). 6 = 7.
a. and other sciences.
w
cube plus three times the quantity a minus
plus 6 multiplied
6.
Read the expressions
of Exs. 33. 6 = 2. 10-14
The
representation of numbers by letters makes it posvery briefly and accurately some of the principles of arithmetic.
.
26. a = 4.12
17
&
*
ELEMENTS OF ALGEBRA
18
'
8
Find the numerical value of 8 a3
21. 6 = 1. 6 = 6.
34. 2-6 of the exercise.
6 = 4.
then
8 = \ V(a + 6 + c) (a 4.
37. Six times a plus 4 times
32.
30.
and
If the three sides of a triangle contain respectively c feet (or other units of length).
27. a
a=3. 6 = 3.
6. a =4.
a = 3. 30.
Twice a3 diminished by 5 times the square root of the quantity a minus 6 square.
a =3.
sible to state
Ex.
The quantity a
6
2 by the quantity a
minus
36.
22.6 . 6 = 5.
Six
2
.
23.
35. 6 = 5. if
:
a = 2. physics.
12 cr6
-f-
6 a6 2
6s. a = 3.
29. a = 2. and the area of the
is
triangle
S
square feet (or squares of other units selected). of this exercise?
What kind of expressions are Exs.
6=2.
:
6. geometry.
Six times the square of a minus three times the cube of Eight x cube minus four x square plus y square. 38.
Express in algebraic symbols 31. a = 4.
28. 6 = 6.
INTRODUCTION
E. if v = 50 meters per second 5000 feet per minute.
the area of the triangle equals
feet. 15 therefore feet. and c
13
and
15
=
=
=
. and 15 feet.
2. A carrier pigeon in 10 minutes.
(b) 5.
S = | V(13-hl4-fl5)(13H-14-15)(T3-14-i-15)(14-13-f-15)
= V42-12-14. Find the height of the tree. 14. c.
By
using the formula
find the area of a triangle
whose
sides are respectively
(a) 3.
i.16 centimeters per second. and 13 inches. A body falling from a state of rest passes in t seconds 2 over a space S (This formula does not take into ac^gt 32 feet.
. b 14.
(c) 4. if v = 30 miles per hour. How far does a body fall from a state of rest in T ^7 of a (c)
A
second ?
3.g.seconds.) Assuming g
. 13. 12. count the resistance of the atmosphere. An electric car in 40 seconds. then a 13. A train in 4 hours.16
1
= 84. d. if v .
84 square
EXERCISE
1. if v
:
a.
9
distance s passed over by a body moving with the uniform velocity v in the time t is represented by the formula
The
Find the distance passed over by A snail in 100 seconds.
4. the three sides of a triangle are respectively 13. and 5 feet.
=
(a)
How
far does
a body fall from a state of rest in 2
seconds ?
(b)
*
stone dropped from the top of a tree reached the ground in 2-J. b.e.
32 F.
(c)
5 miles.
(c)
8000 miles.14d (square units).
If the diameter of a sphere equals d units of length.
2 inches.
If the
(b) 1 inch.
diameter of a sphere equals d
feet.
(c)
5
F.
fo
If
i
represents the simple interest of
i
p
dollars at r
in
n
years.
then the
volume
V=
(a) 10 feet.
square units (square inches.
(c)
10
feet.). This number cannot be expressed exactly.
$ = 3.
~
7n
cubic feet.
of this formula
:
The The
interest on interest
$800
for 4 years at
ty%.
to Centigrade readings:
(b)
Change the following readings
(a)
122 F.
denotes the number of degrees of temperature indi8.
. and the value given above is only an
surface
$=
2
approximation.). the equivalent reading C on the Centigrade scale may be found by the formula
F
C
y
= f(F-32). (The number 3.
5.14
4.
on $ 500 for 2 years at 4 %.) Find the surface of a sphere whose diameter equals
(a)
7. then
=p
n
*
r
%>
or
Find by means
(a)
(b)
6.
:
8000 miles.14 is frequently denoted by the Greek letter TT.
ELEMENTS OF ALGEBRA
If
the radius of a circle
etc.
the area
etc.
6
Find the volume of a sphere whose diameter equals:
(b)
3
feet. the
3.
is
H
2
units of length (inches. Find the area of a circle whose radius is
It
(b)
(a) 10 meters. If cated on the Fahrenheit scale.14
square meters.
meters.
however. we define the sum of two numbers in such a way that these results become general. in algebra this word includes also the results obtained by adding negative.
While
in arithmetic the
word sum
refers only to
the
result obtained
by adding positive numbers. or that
and
(+6) + (+4) = +
16
10. In arithmetic we add a gain of $ 6 and a gain of $ 4.$6) + (-
$4) = (-
$10).
. we call the aggregate value of a gain of 6 and a loss of 4 the sum of the two. SUBTRACTION. but we cannot add a gain of $0 and a loss of $4.
Since similar operations with different units always produce analogous results. the fact that a loss of
loss of
+ $2.
of
$6 and a gain
$4
equals a
$2 may be
represented thus
In a corresponding manner we have for a loss of $6 and a
of
loss
$4
(. or positive and negative numbers. AND PARENTHESES
ADDITION OF MONOMIALS
31. Or in the symbols of algebra
$4) =
Similarly. In algebra. Thus a gain of $ 2 is considered the sum of a gain of $ 6 and a loss of $ 4.CHAPTER
II
ADDITION.
find the numerical values of a + b
-f c-j-c?.
Thus. '.
23.
(always) prefix the sign of the greater.
6
6
= 3.16
32. c =
4.
18.
-
0.
+ (-9).
22. if :
a
a
= 2. 4.
l-f(-2). subtract their absolute values and
. d = 0.
5.
(_
In Exs. 10.
is 0. c = = 5.
d = 5. 23-26. add their absolute values if they have opposite signs.
19. the average of 4 and 8
The average The average
of 2.3.
of:
20.
ELEMENTS OF ALGEBRA
These considerations lead to the following principle
:
If two numbers have the same sign. + -12. 12. the one third their sum. = 5.
33. and the sum of the numbers divided by n. is 2.
4
is
3 J. 5.
EXERCISE
Find the sum
of:
10
Find the values
17.
of 2.
.
21.
(-17)
15
+ (-14).
The average
of two
numbers
is
average of three numbers average of n numbers is the
is one half their sum. 24.
.
42.
= 22.
:
and
1.
6. c = 0. 72.
13. and 3 a. .
sets of
numbers:
13.
31. and $4500 gain. .
.
40.
'
Find the average of the following
34.13. 66.
.
2.
Similar or like terms are terms which have the same
literal factors..5. -11 (Centigrade).4. Find the average temperature of New York by taking the average of the following monthly averages 30.
are similar terms.
d=
3. 37. 55. 10. 6.
which are not
similar. = -13. $7000 gain.ADDITION. Find the average gain per year of a merchant. c=14. 38.
25. 39.5. 0.
. . 12. -4.
36.
= -23. 1. 7 yards. $3000 gain.
AND PARENTHESES
d = l.7.
30. and 3 a.
27. 41.
33.
Dissimilar or unlike terms are terms
4 a2 6c and o
4 a2 6c2 are dissimilar terms. and 3 yards. SUBTRACTION. 32.
and
-8
F.. 09.
What number must be added to 9 to give 12? What number must be added to 12 to give 9 ? What number must be added to 3 to give 6 ? C* What number must be added to 3 to give 6? **j Add 2 yards.
:
48. $1000 loss. 60.7. 32.
-'
1?
a
26. and 3 F.
or 16
Va + b
and
2Vo"+~&. 10.
5 a2 &
6 ax^y and
7 ax'2 y. $500 loss.
35.3. or
and
. 5 and
12. \\ Add 2 a. 10. 74.
2.
:
34. 3. & = 15.
3 and 25. 7 a.
:
Find the average temperature of Irkutsk by taking the average of the following monthly temperatures 12.
&
28.
.
Find the average of the following temperatures 27 F. affected
by the same exponents.
. 6. 7 a. 34.
^
'
37. if his yearly gain or loss during 6 years was $ 5000 gain. and 4. }/ Add 2 a.
29.
4
F. 43.
sum of two such terms can only be them with the -f.
2 a&.
In algebra the word sum is used in a 36. or
a
6. either the difference of a and b or the sum of a and
The sum
of
a.
10.
12
2 wp2 .
14
.
12
13
b sx
xY xY 7 #y
7.
2(a-f &).
Vm
-f.
11.
1
\
-f-
7 a 2 frc
Find the sum of
9. 7 rap2.
5l
3(a-f-6).ii.
:
2 a2.
11
-2 a +3a -4o
2. in algebra it may be considered b.
5 a2
.18
35.
5Vm + w.
2
.
ab
7
c
2
dn
6.
12(a-f b)
12.sign.
The sum The sum
of a of a
Dissimilar terms cannot be united into a single term.
12Vm-f-n. While in arithmetic a denotes a difference only.
EXERCISE
Add:
1.13 rap
25 rap 2.
-3a
.
13. The indicated by connecting and a 2 and
a
is
is
-f-
a2
.
9(a-f-6).
. Algebraic sum.
b
a
-f (
6). b wider sense than in arithmetic.
The sum
x 2 and
f
x2 .
+ 6 af
.
-f
4 a2. and 4 ac2
is
a
2 a&
-|-
4 ac2.
ELEMENTS OF ALGEBRA
The sum
of 3
of
two similar terms
x2
is
is
another similar term.
ADDITION.
3 gives 5
is
evidently 8. ing the sign of the subtrahend thus to subtract 6 a 2 6 and 8 a 2 6 and find the sum of change mentally the sign of
. two numbers are given.2. and the
required number the difference.
1.
From
5 subtract
to
The number which added
Hence.
3. the given number the subtrahend. Or in symbols.
The
results of the preceding examples could be obtained
by the following
Principle. the algebraic sum and one of the two numbers is
The algebraic sum is given. from What 3.
From
5 subtract
to
.
This gives by the same method. In subtraction. a-b =
x.
To
subtract.
5
is
2.
(-
6)
-(-
= . In addition. Therefore any example in subtraction
different
.
From
5 subtract
+ 3.
State the other practical examples which show that the number is equal to the addition of a
40. and their algebraic sum is required.
Ex.
41.
2. called the minvend.3.
7.
a. the other number is required.
.
AND PARENTHESES
23
subtraction of a negative
positive number.
Ex. change the sign
of the subtrahend and
add.
may
be stated in a
:
5 take form e. SUBTRACTION.
if
x
Ex.
NOTE.g.
3 gives
3)
The number which added
Hence. Subtraction is the inverse of addition.
6
-(-3) = 8. The student should perform mentally the operation of chang8 2 6 from 6 a 2 fc. may be stated number added to 3 will give 5? To subtract from a the number b means to find the number which added to b gives a.
+b
3.
changed. one occurring within the other.
AND PARENTHESES
27
SIGNS OF AGGREGATION
43. I.a
-f-
= 4a
sss
7a
12
06
6.
A
moved
w
may be resign of aggregation preceded by the sign inserted provided the sign of evei'y term inclosed is
E.b c = a a
&
-f-
-f.c.
6
o+(
a
+ c) = a =a 6 c) ( 4-.a^6)]
-
}
. If we wish to remove several signs of aggregation.a~^~6)]} = 4 a -{7 a 6 b -[.g.
& -f
c.ADDITION.
a+(b-c) = a +b .
46.
If there is no sign before the first term within a paren*
-f-
thesis.
. Simplify 4 a f
+ 5&)-[-6& +(-25. SUBTRACTION.
tractions
By using the signs of aggregation.
66
2&-a + 6
4a
Answer. may be written as follows:
a
-f ( 4.c.
45. we may begin either at the innermost or outermost. Ex.
II.
4a-{(7a + 6&)-[-6&-f(-2&.6 b -f (.& c
additions
and sub-
+ d) = a + b
c
+ d.
(b
c)
a
=a
6 4-
c. the sign
is
understood.2 b . The beginner will find it most convenient
at every step to
remove only those parentheses which contain
(7 a
no others.
Hence
the
it is
sign
may
obvious that parentheses preceded by the -f or be removed or inserted according to the fol:
lowing principles
44. A sign of aggregation preceded by the sign -f may be removed or inserted without changing the sign of any term.
The minuend is always the of the two numbers mentioned.
In each of the following expressions inclose the last three in a parenthesis preceded by the minus sign
:
-27i2 -3^ 2 + 4r/.
p + q + r-s.
'
NOTE.
The
difference of a
and
6.
4.
of the cubes of
m and
n.
13.
3.
and the subtrahend the second.
Nine times the square of the sum of a and by the product of a and b.
10.
EXERCISES
IN"
ALGEBRAIC EXPRESSION
17
:
EXERCISE
Write the following expressions
I.
y
-f-
8
.
terms
5.
5.
EXERCISE
AND PARENTHESES
16
29
In each of the following expressions inclose the last three terms in a parenthesis
:
1.
3.
The square of the difference of a and b.
Three times the product of the squares of The cube of the product of m and n.
a-\-l>
>
c
+
d.
The sum^)f
m
and
n.
difference of the cubes of n and m.
II.2 tf . SUBTRACTION.
2m-n + 2q-3t.7-fa.
first.
7.
6 diminished
.
5^2
_ r .
8.
5 a2
2.
6.1. 9. )X
6.
m and n.
The product The product
m
and
n.
The sum
of tKe squares of a
and
b. The product of the sum and the difference
of
m and n.
4 xy
7 x* 4-9 x + 2.
m
x
2
4.
12.
2. .
7.4 y* .
The sum
of the fourth powers of a of
and
6.
z
+ d.ADDITION.
The The
difference of the cubes of
m
and
n.
6.
ELEMENTS OF ALGEBRA
The sum
x.)
.
x cube minus quantity 2 x2 minus 6 x plus
The sum
of the cubes of a.
a plus the prod-
uct of a and
s
plus the square of
-19. 16.
d.30
14.
18.
b.
and
c
divided by the
ference of a and
Write algebraically the following statements:
V 17.
The sum
The
of a
and
b multiplied
b is equal to the difference of
by the difference of a and a 2 and b 2
.
dif-
of the squares of
a and
b increased
by the
square root of
15.
6 is equal to the square of
b. The difference of the squares of two numbers divided by the difference of the numbers is equal to the sum of the two numbers. (Let a and b represent the numbers.
difference of the cubes of a
and
b divided
by the
difference of
a and
6.
therefore. what force is produced by the addition of 5 weights at B ? What.CHAPTER
III
MULTIPLICATION
MULTIPLICATION OF ALGEBRAIC NUMBERS
EXERCISE
18
In the annexed diagram of a balance. If the two loads balance.
3.
If the
two loads balance. weights at A ? Express this as a multibalance. weight at A ? What is the sign of a 3 Ib. let us consider the and JB. and forces produced at by 3 Ib.
5. 2.
4. is 5 x ( 3) ?
7.
force is produced
therefore. weight at B ?
If the
addition of five 3
plication example. applied at let us indicate a downward pull at by a positive sign. what force is produced by the Ib.
If the two loads
what
What.
two loads balance.
A
A
A
1. is
by
taking away 5 weights from
A?
5
X 3?
6.
By what sign is an upward pull at A represented ? What is the sign of a 3 Ib. what force
31
is
produced by tak(
ing away 5 weights from
B ? What therefore is
5)
x(
3) ?
. weights.
becomes meaningless
if
definition. Multiplication by a positive integer is a repeated addition.
however.
examples were generally
method of the preceding what would be the values of
(
5x4.32
8. To take a number 7 times.
48. and we may choose any definition that does not lead to contradictions.
x
11.
ELEMENTS OF ALGEBRA
If
the signs obtained by the
true.4) x
braic laws for negative
~ 3> = -(. make venient to accept the following definition
:
con-
49.
4
x(-8) = ~(4)-(4)-(4)=:-12. a result that would not be obtained by other assumptions.
the multiplier is a negative number.
. 5x(-4).
NOTE. 4 multiplied by 3. times is just as meaningless as to fire a gun
tion
7
Consequently we have to define the meaning of a multiplicaif the multiplier is negative.
Thus.
Multiplication
by a negative
integer is a repeated
sub-
traction. 9 x
(-
11). This definition has the additional advantage of leading to algenumbers which are identical with those for positive numbers. or plied by 3. 4 multi44-44-4 12.9) x
11.
9
9.
In multiplying integers we have therefore four cases
trated
illus-
by the following examples
:
4x3 = 4-12.
(-
9)
x (-
11) ?
State a rule by which the sign of the product of
two
fac-
tors can be obtained. (-5)X4.4)-(.
(
(.
(. or
4x3 =
=
(_4) X
The preceding
3=(-4)+(-4)+(-4)=-12.
4x(-3)=-12.4)-(-4) = +
12. such as given in the preceding exercise. thus. Practical examples^
it
however.
Multiply 2 a .3 a 2 + a8 . If the polynomials to be multiplied contain several powers of the same letter.a6 4 a 8 + 5 a* . 1 being the most convenient value to be substituted for all letters.3
b
by a
5
b. Since errors.
Ex. multiply each term of one by each term of the other and add the partial products thus formed.
The most convenient way of adding the partial products is to place similar terms in columns.
59.a
.
. are far more likely to occur in the coefficients than anywhere else.
2a-3b a-66 2 a .
If
Arranging according to ascending powers
2
a
.3 ab
2
2 a2
10 ab
-
13 ab
+ 15 6 2 + 15 6 2
Product.
2.a6
=2
by numerical
Examples
in multiplication can be checked
substitution.3 a
3
2
by 2 a
:
a2 + l. as illustrated in the following example
:
Ex.2 a2 6 a8
2 a*
*
-
2"
a2
-7
60. however. the student should
apply this test to every example.3 a 2 + a8
a
a = =-
I
1
=2
-f
2
a
4.
Check. the work becomes simpler and more symmetrical by arranging these expressions according to either ascending or descending powers.1.
a2
+ a8 + 3 .M UL TIP LICA TION
37
58. To multiply two polynomials. Since all powers of 1 are 1.4. this method
tests only the values of the coefficients
and not the values of
the exponents.
Multiply 2
+ a -a.
The square
2
(a 4. plus the product of the
EXERCISE
Multiply by inspection
1. (4s + y)(3-2y).
6. plus the
last terms.42
ELEMENTS OF ALGEBRA
of the result is obtained
product of 5 x
follows:
by adding the These products are frequently called the cross products.
.
2
(2m-3)(3m + 2).
The middle term
or
Wxy-12xy
Hence in general.
3. 9.
4.
that the square of each term is while the product of the terms may have plus always positive.
(2a-3)(a + 2).
2
2
+ 2) (10 4-3). 13. (100 + 3)(100 + 4).
(x
i-
5
2
ft
x 2 -3 6 s). the product of two binomials whose corresponding terms are similar is equal to the product of the first two
terms.
2
(2x y
(6
2
2
+ z )(ary + 2z ).
((5a?
(10
12.
2 2 2 2 (2a 6 -7)(a & +
5).
2
10. 14. and are represented as
2 y and 4y 3 x.
7%e square of a polynomial is equal to the sum of the squares of each term increased by twice the product of each term with each that follows it. ) (2
of a polynomial.
5. (3m + 2)(m-l).
65.
7.
11.
or
The student should note
minus
signs.
8.
sum of the
cross products.
(5a-4)(4a-l).
(5a6-4)(5a&-3).-f
2 a&
-f
2 ac
+ 2 &c.&
+ c) = a + tf + c
.
:
25
2.
is the process of finding one of two factors and the other factor are given. The dividend is the product of the two factors, the divisor the given factor, and the quotient is the required factor.
67.
Division
if
their product
is
Thus
by
-f
to divide
12.
12
by
+
3,
we must find
is
the
;
number which
3 gives
But
this
number
4
hence
_
multiplied
12 r +3
=4.
68.
Since
-f
a
-
-f b
-fa
_a
and
it
-f-
a
= -f ab = ab b = ab b = ab,
b
-f-
follows that
4-a
=+b
ab
a
ab
a
69.
Hence the law
:
of signs
is
the same in division as in
multiplication
70.
Like signs produce plus, unlike signs minus.
Law
of
,
a8 -5- a5
=a
3
for a 3
It follows from the definition that Exponents. X a5 a8
=
.
Or
in general, if
greater than
m n, a
-f-
and n are positive integers, and m ~ n an = a m a" = a'"-", for a
<
m
m
is
45
46
ELEMENTS OF ALGEBRA
71. TJie exponent of a quotient of two powers with equal bases equals the exponent of the dividend diminished by the exponent
of the divisor.
DIVISION OF MONOMIALS
7 3 72. To divide 10x y z by number which multiplied by number is evidently
2x y
6
2
,
we have
z
to
find
the
2x*y
gives 10 x^ifz.
This
Therefore,
the quotient
*
,
= - 5 a*yz.
is
Hence,
sign,
of two monomials of their
part
coefficients,
is the
a monomial whose
coefficient is the quotient
preceded by the proper
literal
and whose
literal
found
in accordance with the
quotient of their law of exponents.
parts
73. In dividing a product of several factors by a number, only one of these factors is divided by that number. Thus (8 12 20)-?-4 equals 2 12 20, or 8 3 20 or 8 12 5.
-
-
.
-
.
-
.
EXERCISE
Perform the divisions indicated
'
:
28
'
2
.
76-H-15.
-39-*- 3.
2
15
3"
7
7'
3.
-4*
'
4.
5.
-j-2
12
.
4
2
9
5 11
68
3 19 -j-3
5
10.
(3
38
-
-2 4 )^(3 4 .2 2).
56
'
11.
3
(2
.3*.5 7 )-f-(
2
'
12
'
2V
14
36 a
'
13
''
y-ffl-g
35
-5.25
-12 a
2abc
15
-42^
'
-56aW
'
UafiV
DIVISION
lg
47
-^1^. 16 w
7
20>
7i
9
_Z^L4L.
22.
10 iy.
132 a V* 14 1
*
01
-240m
120m-
40
6c
fl
/5i.
3J)
c
23.
2 (15- 25. a ) -=- 5.
25. 26.
(18
(
.
5
.
2a )-f-9a.
2
24.
(7- 26 a
2
)
-f-
13.
DIVISION OF POLYNOMIALS BY MONOMIALS
To divide ax-}- fr.e-f ex by x we must find an expression which multiplied by x gives the product ax + bx -J- ex.
74.
But
TT
x(a
aa?
Hence
+ b e) ax + bx + ex. + bx -f ex = a 4- b +
-\.
,
.
c.
a?
To divide a polynomial by a monomial, cfc'wde each term of the dividend by the monomial and add the partial quotients thus
formed.
3 xyz
EXERCISE
Perform the operations indicated
1.
:
29
2.
5.
fl
o.
(5*
_5* + 52)
-5.
52
.
3.
97
.
(2
(G^-G^-G^-i-G
(11- 2
4.
(8- 3
+
11 -3
+ 11
-5)-*- 11.
18 aft- 27 oc
Q y.
9a
4
-25 -2 )^-2
<?
2
.
+8- 5 + 8-
7) -*-8.
5a5 +4as -2a
2
-a
-14gV+21gy
Itf
15 a*b
-
12
aW + 9 a
2
2
3a
48
,
ELEMENTS OF ALGEBRA
22
4,
m n - 33 m n
4
s
2
-f
55
mV
- 39 afyV + 26 arVz 3
- 49 aW + 28 a -W - 14 g 6 c
4 4
15. 16.
2 (115 afy -f 161 afy
- 69
4
2
a;
4
?/
3
- 23 ofy
3
4
)
-5-
23 x2y.
(52
afyV - 39
4
?/
oryz
- 65 zyz - 26 tf#z)
-5-
13 xyz.
-f-
,
17.
(85 tf
- 68 x + 51 afy - 34 xy* -f 1 7
a;/)
- 17
as.
DIVISION OF A POLYNOMIAL BY A POLYNOMIAL
75.
Let
it
be required to divide 25 a
- 12 -f 6 a - 20 a
3
2
by
2 a 2 -f 3 a, divide
4
a, or, arranging according to
2
descending powers of
6a3 -20a
-f
25a-12
2 by 2a -
The term containing the highest power of a in the dividend (i.e. a 8 ) is evidently the product of the terms containing respectively the highest power of a in the divisor and in the quotient.
Hence the term containing the highest power
of a in the quotient is
If
the product of 3 a and 2
2
4 a
+
3, i.e.
6 a3
12 a 2
-f
9 a, be sub-
8 a 2 -f 16 a tracted from the dividend, the remainder is 12. This remainder obviously must be the product of the divisor and the rest of the quotient. To obtain the other terms of the quotient we have
therefore to divide the remainder,
8 a2
-f-
16 a
12,
2 by 2 a
4 a
+
3.
consequently repeat the process. By dividing the highest term in the new dividend 8 a 2 by the highest term in the divisor 2 a 2 we obtain
,
We
4,
the next highest term in the quotient. 4 by the divisor 2 a2 4 a Multiplying
-I-
+ 3, we
obtain the product
8 a2
16 a
12,
which subtracted from the preceding dividend leaves
the required quotient.
no remainder. Hence 3 a
4
is
DIVISION
The work
is
49
:
usually arranged as follows
- 20 * 2 + 3 0a-- 12 a 2 +
a3
25 a
{)
-
12
I
2 a2 8 a
-
4 a 4
a
_
12
+3
I
-
8 a? 4- 16
a-
76. The method which was applied in the preceding example may be stated as follows 1. Arrange dividend and divisor according to ascending or
:
descending powers of a common letter. 2. Divide the first term of the dividend by the first term of the divisor, and write the result for the first term of the quotient.
3.
Multiply this term of the quotient by the whole divisor, and
subtract the result
4.
from
it
the dividend.
the same order as the given new dividend, and proceed as before.
Arrange
the
remainder in
as a
expression, consider
5.
until the highest poiver
Continue the process until a remainder zero is obtained, or of the letter according to which the dividend
is less
was arranged
the divisor.
than the highest poiver of the same
letter in
77.
Checks.
Numerical substitution constitutes a very con-
venient, but not absolutely reliable check. An absolute check consists in multiplying quotient and divisor. The result must equal the dividend if the division
was
exact, or the dividend diminished by the remainder division was not exact.
The
first
member
or left side of an equation
is
that part
The secof the equation which precedes the sign of equality. y y or z) from its relation to
63
An
known numbers.
hence
it
is
an equation
of
condition.
Thus.CHAPTER V
LINEAR EQUATIONS AND PROBLEMS
79. An equation of condition is usually called an equation. second member is x
+
4
x
9. in the equation 2 x 0.
82.
which
is
true for all values
a2 6 2 no matter what values we assign to a Thus.
. An equation of condition is an equation which is true only for certain values of the letters involved.
the
80.
83.
x
20. ond member or right side is that part which follows the sign of
equality. An identity is an equation of the letters involved.
in
Thus x
12 satisfies the equation x
+
1
13.
y
=
7 satisfy
the equation x
y
=
13.
.r
-f9
= 20
is
true only
when
a.
.
ber
equation is employed to discover an unknown num(frequently denoted by x.
the
first
member
is
2 x
+
4.
A set of numbers which when substituted for the letters an equation produce equal values of the two members. The sign of identity sometimes used is = thus we may write
.
(rt+6)(a-ft)
=
2
-
b'
2
. (a + ft) (a b) and b. is said to satisfy an equation.
81.
=11.
A
linear equation or
which when reduced
first
to its simplest
an equation of the first degree is one form contains only the
as
9ie
power of the unknown quantity.
To
solve
an equation
to find its roots.b.
If equals be divided by equals. the products are equal.
2.
ELEMENTS OF ALGEBRA
If
value of the
an equation contains only one unknown quantity.
4.g. x
I. an^ unknown quantity which satisfies the equation is
a root of the equation.
86.
If equals be subtracted from equals.
2
= 6#-f7.e.
85.
The process
of solving equations depends upon the
:
lowing principles.
Axiom
4
is
not true
if
0x4
= 0x5. 87. A
2
a.54
84.
If equals be multiplied by equals.
expressed in arithmetical numbers
literal
is
as (7
equation
is
one in which at least one of the
known
quantities as x -f a letters
88. called axioms
1.
A
term may be transposed from
its sign.
.2.
A numerical
equation is one in
which
all
.
NOTE. Consider the equation b Subtracting a from both members.
90.
but 4 does not equal
5.
Like powers or
like roots
of equals are equal.
one member to another by changing
x + a=.
.
89.
the
known quan
x) (x -f 4)
tities are
=
.
a.
3.
fol-
A
linear equation is also called a simple equation.
E.
9
is
a root of the equation 2 y
+2=
is
20.
If equals be added
to equals.
(Axiom
2)
the term a has been transposed from the left to thQ
right
member by changing
its
sign.
= bx
expressed by a letter or a combination of
c. the quotients are equal.
5. the remainders are
equal.
the divisor equals zero.
Transposition of terms. the
sums are
equal.
3. is b. greater one is g.
4. Hence 6 a must be added
to a to give
5.
EXERCISE
1.
two numbers
and the and the
2
Find the greater one.
$> 100 yards cost one hundred dollars. 15.
9. or 12 7.
5. Divide 100 into two
12.
Find the greater one. so that one part
The
difference between
is s.58
Ex. The difference between two numbers Find the smaller one.
Ex.
10.
33
2.
If 7
2.
smaller one
16.
By how much does a exceed 10 ? By how much does 9 exceed x ? What number exceeds a by 4 ? What number exceeds m by n ? What is the 5th part of n ? What is the nth part of x ? By how much does 10 exceed the third part of a? By how much does the fourth part of x exceed b ? By how much does the double of b exceed one half Two numbers differ by 7. is d.
Divide a into two parts.
14.
6. so that
of c ?
is
p.
x
-f-
y yards cost $ 100
.
7. one yard will cost -
Hence
if
x
-f
y yards cost $ 100.
one part equals
is 10. 11.
is
a?
2
is
c?.
17.
1.
find the cost of one yard. 6. one yard will cost
100
-dollars. so that one part Divide a into two parts.
a. and the smaller one
parts.
What number divided by 3 will give the quotient a? ? What is the dividend if the divisor is 7 and the quotient
?
.
ELEMENTS OF ALGEBRA
What must
be added to a to produce a sum b ?
:
Consider the arithmetical question duce the sum of 12 ?
What must
be added to 7 to pro-
The answer is 5.
13.
rectangular field is x feet long and the length of a fence surrounding the field.
is
A A
is
# years
old.LINEAR EQUATIONS AND PROBLEMS
18.
If
B
gave
A
6
25. 28.
What What What What
is
the cost of 10 apples at x cents each ?
is
is is
x apples cost 20 cents ? the price of 12 apples if x apples cost 20 cents ? the price of 3 apples if x apples cost n cents ?
the cost of 1 apple
if
.
20.
y years
How
old was he 5 years ago ?
How
old will he be 10 years hence ?
23.
34.
How many
cents
has he ?
27.
numbers
is x.
Find
21.
22.
How many
cents had he left ?
28.
feet wider than the one
mentioned in Ex. find the
has ra dollars.
How many
cents are in d dollars ? in x dimes ?
A has
a
dollars.
24.
59
What must
The
be subtracted from 2 b to give a?
is a.
How many years
A
older than
is
B?
old.
A
feet wide.
Find the area of the Find the area of the
feet
floor of
a room that
is
and 3
30.
sum
If A's age is x years.
19.
and 4
floor of a room that is 3 feet shorter wider than the one mentioned in Ex.
A man
had a
dollars. A room is x feet long and y feet wide. 28.
?/
31. amount each will then have. and B has n dollars.
26.
Find
35. b dimes. find the of their ages 6 years hence. square feet are there in the area of the floor ?
How many
2 feet longer
29. Find the sum of their ages
5 years ago.
A
dollars.
and
c cents.
and
B
is
y years old.
smallest of three consecutive numbers
Find
the other two. 33. The greatest of three consecutive the other two.
32.
and spent
5 cents. and B's age is y years.
The numerator
If
of a fraction exceeds the denominator
by
3. and "by as much as" Hence we have means equals (=)
95." we have to consider that in this by statement "exceeds" means minus ( ).
a.60
ELEMENTS OF ALGEBRA
wil\
36.
b
To express in algebraic symbols the sentence: " a exceeds much as b exceeds 9.
Find the
number. The first pipe x minutes. and the second pipe alone fills it in
filled
y minutes.
Find
a.
-46.
Find x
% %
of 1000.
How many
x years ago
miles does a train
move
in
t
hours at the
rate of x miles per hour ?
41.
48.
If a
man walks
?
r miles per hour.
% % %
of 100
of
x.
how many
how many
miles will
he walk in n hours
38. how many miles he walk in n hours ?
37.
49.
The two
digits of a
number
are x and
y.
-.
miles does
will
If a man walks r miles per hour.
of m.
of 4. he walk each hour ?
39.
as
a exceeds
b
by as much
as c exceeds 9. Find a
47. find the fraction. If a man walks n miles in 4 hours.
A
was 20 years
old.
A
cistern can be filled
in
alone
fills it
by two pipes.
How
old
is
he
now ?
by a pipe in x minutes. What fraction of the cistern will be second by the two pipes together ?
44.
per
Find 5 Find 6
45.
A
cistern
is
filled
43.
c
a
b
=
-
9. If a man walks 3 miles per hour.
.50. What fraction of the cistern will be filled by one pipe in one minute ?
42. in how many hours he walk n miles ?
40.
m is the
denominator.
thus:
a
b
= c may
be expressed as follows
difference between a
:
The
and
b is
c.
8
-b ) + 80 = a
.
=
2
2
a3
(a
-
80.
by one third of b equals 100.
9.
4.
The double
as
7.
3. 2.
6.
equal to the
sum and the difference of a and b sum of the squares of a and
gives the
Twenty subtracted from 2 a
a. the difference of the squares of a
61
and
b increased
-}-
a2
i<5
-
b'
2
'
by 80 equals the excess of a over
80
Or.
5.
of a increased
much
8.
a exceeds b by c.
double of a
is
10.
In
many
word
There are usually several different ways of expressing a symbolical statement in words.
of x increased by 10 equals
x.
of a and 10 equals 2
c.
same
result as 7
subtracted from
. Four times the difference of a and b exceeds c by as
d exceeds
9. a is greater than b by b is smaller than a by
c.
cases it is possible to translate a sentence word by in algebraic symbols in other cases the sentence has to be changed to obtain the symbols.
The product
of the
is
diminished by 90 b divided by 7.
EXERCISE
The The double The sum
One
34
:
Express the following sentences as equations
1.LINEAR EQUATIONS AND PROBLEMS
Similarly.
-80.
The
excess of a over b
is c.
c.
c.
third of x equals
difference of x
The
and y increased by 7 equals
a.
80. etc.
pays to C $100.
(c)
If each
man
gains $500.
A
If
and
B
B together have $ 200 less than C..*(/)
(g)
(Ji)
Three years ago the sum of A's and B's ages was 50.
. the first sum exceeds b % of the second sum by
first
(e)
%
of the first plus 5
%
of the second plus 6
%
of the
third
sum
equals $8000.
and C's age
4
a. of 30 dollars. 16.
A
gains
$20 and B
loses
$40.
17.
18. the first sum equals 6 % of the third sura.
x
is
100
x%
is
of 700. 3 1200 dollars. express in algebraic symbols
:
-700.
11.
->.
(a)
(b)
(c)
A is twice as old as B.
as 17
is
is
above
a.
first
00
x % of the
equals one tenth of the third sum.
amounts.
6
%
of m.
12. (d) In 10 years A will be n years old.
50
is
x % of
15. express in algebraic
3x
:
10. a second sum. 14. B's.
m is x %
of n. the
sum
and C's
money
(d)
(e)
will be $ 12.
#is5%of450.
ELEMENTS OF ALGEBRA
Nine
is
as
much below a
13.
x
4-
If A. they have equal
of A's. and C's ages will be 100.
a. a third sum of 2 x + 1 dollars. In 3 years A will be twice as old as B.62
10. B. (e) In 3 years A will be as old as B is now. sum equals $20. they have equal amounts. Express as
:
equations of the (a) 5
(b)
(c)
% a%
of the second
(d)
x c of / a % of
4
sum equals $ 90.
is
If A's age is 2 x. and C have respectively 2 a.000. A is 4 years older than
Five years ago A was x years old. In 10 years the sum of A's. B's. and
(a)
(6)
A
If
has $ 5 more than B. B's age
20.
symbols
B.
5x
A sum of money consists of x dollars.
a.
by 20
40 exceeds 20 by 20. In order to solve them.
Ex.
Simplifying.
equation is the sentence written in alyebraic shorthand.
x+16 = 3(3-5). x = 20. x + 15 = 3 x
3x 16
15.
Dividing.
.
6 years ago he was 10
.
number. Write the sentence in algebraic symbols.
NOTE.
-23 =-30. = x x
3x
-40
3x
40-
Or. the
.
Check.
but
30
=3
x
years. be three times as old as he was 5 years ago.
In 15 years
10. The equation can frequently be written by translating the sentence word by word into algebraic symbols in fact.LINEAR EQUATIONS AND PROBLEMS
63
PROBLEMS LEADING TO SIMPLE EQUATIONS
The simplest kind of problems contain only one unknown number. verbal statement (1)
(1) In 15 years
A
will
may be expressed in symbols (2). the required
.
3z-40:r:40-z. x= 15. 4 x = 80.
3 x or 60 exceeds 40
+ x = 40 + 40.
Ex. 2. etc.
Uniting. In 15 years A will be three times as old as he was 5 years ago.
1.
Uniting. The student should note that x stands for the number of and similarly in other examples for number of dollars. The solution of the equation (jives the value of the unknown number.
Let x
The
(2)
= A's present age.
Transposing.
Transposing.
A
will
Check.
much
as 40 exceeds the number.
3
x
+
16
=
x
x
(x
-
p)
Or.
Let x = the number.
be 30
.
Three times a certain number exceeds 40 by as Find the number.
number by x (or another letter) and express the yiven sentence as an equation. 15. denote the unknown
96. Three times a certain no. number of
yards. Find A's present age. exceeds 40 by as much as 40 exceeds the no.
2.
Uldbe
66
| x x
5(5 is
=
-*-.
Dividing. How long is the Suez
Canal?
10.64
Ex. Find the number.
47 diminished by three times a certain number equals 2.
EXERCISE
1.
A will
be three times as old as to-da3r
. 11.
14 50
is
is
4
what per cent of 500 ? % of what number?
is
12.
4. by as much as 135 ft.
35
What number added
to twice itself gives a
sum
of
39?
44.
.
Let x
3.
120.
3. then the
problem expressed in symbols
W
or.
Find the width of the Brooklyn Bridge.
Hence
40
= 46f.
Find the number whose double increased by 14 equals Find the number whose double exceeds 40 by 10.
What number
7
%
of
350?
Ten times the width of the Brooklyn Bridge exceeds 800 ft.
Find the number.
Find
8. A train moving at uniform rate runs in 5 hours 90 miles more than in 2 hours.
ELEMENTS OF ALGEBRA
56
is
what per cent
of 120 ?
= number
of per cent. exceeds the width of the bridge. 14.
How
old
is
man will be he now ?
twice as old as he was
9.
300
56.
A
number added
number.
Forty years hence
his present age.
Find the number whose double exceeds 30 by as much
as 24 exceeds the number.
13. % of
120. Four times the length of the Suez Canal exceeds 180 miles by twice the length of the canal.
Six years hence a
12 years ago.
twice the number plus
7.
5.
to
42 gives a
sum
equal to 7 times the
original
6. How many miles per hour does it run ?
.
1.
make A's money
equal to 4 times B's
money
wishes to purchase a farm containing a certain He found one farm which contained 30 acres too many. two verbal statements must be given. written in algebraic
symbols.
65
A
and
B
$200.
The problem consists of two statements I.
x. B will have lars has A now?
17.
How many
dol-
A has
A
to
$40. If the first farm contained twice as many acres as
A man
number
of acres. If a problem contains two unknown quantities.
One number exceeds another by
:
and their sum
is
Find the numbers.
14. The sum of the two numbers is 14. which gives the value of
8. In 1800 the population of Maine equaled that of Vermont.
A
and
B
have equal amounts of money.
is
the equation.
Ex. If A gains A have three times as much
16.
numbers (usually the smaller one) by
and use one of the
given verbal statements to express the other unknown number in terms of x.
. Maine's population increased by 510. while in the more complex probWe denote one of the unknown
x.
times as
much
as A. Vermont's population increased by 180.000.
statements are given directly.000.
F
8. then dollars has each ? many
have equal amounts of money.
How many dollars
must
?
B
give to
18. and
as
15.
how many
acres did he wish to
buy
?
19. Find
the population of Maine in 1800. Ill the simpler examples these two
lems they are only implied. and another which lacked 25 acres of the required number.
the second one.
five
If
A
gives
B
$200. The other verbal statement.
B
How
will loses $100. One number exceeds the other one by II. and Maine had then twice as many inhabitants as Vermont.
97. and
B
has $00. During the following 90 years.LINEAR EQUATIONS AND PROBLEMS
15.
to
Use the simpler statement.
A will lose.
/
. the smaller number.
which leads
ot
Ex.
Let
x
14
I
the smaller number. although in general the simpler one should be selected.
. 8 = 11.
x
= 8.
terms of the other.
2x
a?
x
-j-
= 6. = A's number of marbles.66
ELEMENTS OF ALGEBRA
Either statement may be used to express one unknown number in terms of the other.
expressed symbols is (14 x) course to the same answer as the first method.
o\
(o?-f 8)
Simplifying.
has three times as many marbles as B.
+
a-
-f
-f
8
= 14.
Statement
x
in
=
the larger number. 8 the greater number.
The two statements
I. B will have twice as
viz.
Another method for solving this problem is to express one unknown quantity in terms of the other by means of statement II viz. A has three times as many marbles as B.
. 26 = A's number of marbles after the exchange. consider that by the
exchange
Hence. I.
= B's number of marbles.
x
x =14
8. 25 marbles to B. unknown quantity
in
Then.= The second statement written
the equation ^
smaller number.
.
If
A gives
are
:
A
If
II. B will have twice as many as A.
<
Transposing. A gives B 25 marbles.
26
= B's number of marbles after the exchange.
Uniting.
in algebraic
-i
symbols produces
#4a. = 14. 2. the sum of the two numbers is 14. the greater number. Let
x
3x
express one
many as A.
Then.
To
express statement II in algebraic symbols.
x
3x
4-
and
B
will gain. = 3.
If
we
select the first one.
and
Let x
= the
Then x -+.
Dividing.
45 . the number of half dollars. Find the numbers.240.
The numbers which appear
in the equation should
always
be expressed in the same denomination.. consisting of half dollars and dimes..5 x . 3. then. the number of dimes. How many are there of each ?
The two statements are I.
6 times the smaller.$3. 40 x .
*
'
.75. Check. x = 6.
Find
the numbers.
Two numbers
the smaller. 3 x = 45. .
Check. The value of the half
:
is 11.
x
from
I. The number of coins II. B's number of marbles. Dividing.
6 dimes
= 60
= 310.
Simplifying. x = the number of half dollars.10.
67
x
-f
25 25
Transposing.25 = 20.
Uniting.
we
express the statement II in algebraic symbols.LINEAR EQUATIONS AND PROBLEMS
Therefore.
by 44. of dollars to the number of cents.
50(11 660 50 x
-)+ 10 x = 310. 6 half dollars = 260 cents..
w'3.
2. cents.
x x
+
= 2(3 x = 6x
25
25).
60. and the Find the numbers. etc.
50. have a value of $3.
Let
11
= the number of dimes. their sum
+ +
10 x 10 x
is
EXERCISE
36
is five
v
v.550 -f 310.
*
98. A's number of marbles.10.
Dividing.
.
(Statement II)
Qx
.
is 70. the
price.
Never add the number number of yards to their
Ex.
differ
differ
and the greater and their sum
times
Two numbers
by
60. 11 x = 5.
greater
is
.
1.10.
Simplifying.
The sum of two numbers is 42.
50 x
Transposing.
Selecting the cent as the denomination (in order to avoid fractions).
Uniting. x = 15. 15 + 25 = 40. but 40 = 2 x 20.
dollars
and dimes
is
$3. Eleven coins.
000
feet. McKinley.
ELEMENTS OF ALGEBRA
One number
is
six
times another number.
On December
21. Mount Everest is 9000 feet higher than Mt. and four times the former equals five times the latter.
find the
weight of a cubic
Divide 20 into two parts.
A's age is four times B's.
cubic foot of iron weighs three times as much as a If 4 cubic feet of aluminum and
Ibs.
as the larger one.
?
Two
vessels contain together 9 pints. and twice the altitude of Mt. one of which increased by
9.
tnree times the smaller by 65. How many inches are in each part ?
15. What are their ages ?
is
A A
much
line 60 inches long is divided into two parts. the larger part exceeds five times the smaller part by 15 inches.
7. Everest by 11. McKinley exceeds the altitude of
Mt.
2 cubic feet of iron weigh 1600 foot of each substance.
and in Mexico
?
A
cubic foot of aluminum.
6. Find their ages.
it
If the smaller
one contained 11 pints more.
How many
14 years older than B.
11.. the number.
How many
hours does the day last ?
.
3 shall be equal to the other increased by
10. Twice 14.
would contain three times as
pints does each contain ?
much
13. the night in
Copenhagen
lasts 10 hours
longer than the day.
of volcanoes in
Mexico exceeds the number
of volcanoes in the United States by 2.
and twice the greater exceeds Find the numbers.
9.
5.68
4.
Find
Find two consecutive numbers whose sum equals 157. How many volcanoes are
in the
8. and B's age is as below 30 as A's age is above 40. and in 5 years A's age will be three times B's.
United
States. and the
greater increased by five times the smaller equals 22.
What
is
the altitude of each
mountain
12.
Two numbers
The number
differ
by
39.
Let
x
II. they would have 3. The solution gives
:
3x
80
Check.LINEAR EQUATIONS AND PROBLEMS
99. original amount.
are
:
C's
The three statements
A. Tf it should be difficult to express the selected verbal state-
ment
directly in algebraical symbols. If A and B each gave $5 to C. let us consider the words ** if A and B each gave $ 5 to C.
has. The third
verbal statement produces the equation. II. and 68. three One of the unknown num-
two are expressed in terms by means of two of the verbal statements.
number
had.
I.
Ex. and B has three as A.
8(8
+ 19)
to C.
number
of dollars of dollars
B
C
had."
To
x
8x
90
= number of dollars A had after giving $5.
times as
much
as
A. B.
A
and B each gave $ 5
respectively. bers is denoted by x. B.
the
the
number
of dollars of dollars of dollars
A
B
C
has. and C together have $80. or 66 exceeds 58 by 8. then
three times the
money by
I. and the other
of x
problem contains three unknown quantities. 4 x = number of dollars C had after receiving $10. number of dollars A had. If A and B each gave $5 to C. = 48. try to obtain
it
by a
series of successive steps.
sum of A's and B's money would exceed much as A had originally. and C together have $80. then three times the sum of A's and B's money would exceed C's money by as much as A had originally.
first
According to
3 x
number
number
and according
to
80
4
x
=
the
express statement III by algebraical symbols. B has three times as much as A.
69
If a
verbal statements must be given. III. x = 8.
1.
. = number of dollars B had after giving $5.
5
5
Expressing in symbols Three times the sum of A's and B's money exceeds C's money by A's 3 x ( x _5 + 3z-5) (90-4z) = x. 19.
has.
If
4x
= 24.
4 x -f 8 = 28. = the number of dollars spent for horses. Let
then. = the number of dollars spent for cows. 9 cows. and the sum of the
.
x
-j-
= the
number of horses. 90
may
be written. 28 2 (9 5). 9 -5 = 4 . 2.
first. and each sheep $ 15.
Dividing.
and. the third five times the first.
28 x 15 or 450
5 horses. each horse costing $ 90.
1 1
Check. cows.
The total cost equals $1185. each cow $ 35.
+
8
90 x
and. x = 5. and Ex. sheep.
III. and the difference between the third and the second is 15
2. = the number of dollars spent for sheep
Hence statement
90 x
Simplifying. The number of cows exceeds the number of horses by 4.
A
and the number of sheep was twice as large as the number How many animals of each kind did he buy ?
of horses and cows together. The number of sheep is equal to twice tho number of horses and
x 4
the
cows together. x -f 4 = 9.140 + (50 x x 120 = 185. 2 (2 x -f 4) or 4 x
Therefore. number of horses.
three statements are
:
IT. number of cows.
The
I.
Uniting. The number of cows exceeded the number of horses by
4.
85 (x 15 (4 x
I
+ 4)
+
8)
= the number of sheep.
x
Transposing. according to III.
x 35
-f
+
=
+
EXERCISE
1.
first
the third exceeds the second by and third is 20. 90 x -f 35 x + GO x = 140 20 + 1185. according to II.
+ 35 (x +-4) -f 15(4z-f 8) = 1185.
Find three numbers such that the second is twice the 2.
37
Find three numbers such that the second is twice the first. and 28 sheep would cost 6 x 90 -f 9 + 316 420 = 1185. number of sheep. 185 a = 925. + 35 x 4.70
ELEMENTS OF ALGEBRA
man spent $1185 in buying horses. number of cows.
what is the length of each?
has 3.
what are the three angles ?
10. the second one is one inch longer than the first.
If the second angle of a triangle is 20 larger than the and the third is 20 more than the sum of the second and
first.
twice the
6.000 more inhabitants than Philaand Berlin has 1. and the pig iron produced in one year (1906) in the United States represented together a value
.
is five
numbers such that the sum of the first two times the first.
twice as old as B. and children together was 37.
-
4. equals 49 inches.
"Find three
is 4.
first.000. the copper. The gold.
the
first
Find three consecutive numbers such that the sum of and twice the last equals 22.000.
A
is
Five years ago the What are their ages ?
C. and the third exceeds the
is
second by
5.
and
of the three sides of a triangle is 28 inches. increased by three times the second side.
7.
v
.LINEAR EQUATIONS AND PROBLEMS
3.000 more than Philadelphia (Census 1905). and 2 more men than women. and the sum of the first and third is 36.
Find three consecutive numbers whose sum equals
63. If twice
The sum
the third side. how many children
were present ?
x
11.
first. and the third part exceeds the second by 10.
the third
2.
13. and is 5 years younger than sum of B's and C's ages was 25 years.
The
three angles of any triangle are together equal to
180. men. In a room there were three times as many children as If the number of women. what is the population of each city ?
8.
71
the
Find three numbers such that the second is 4 less than the third is three times the second.
9.
New York
delphia.
A
12.
v
-
Divide 25 into three parts such that the second part first. women. If the population of New York is twice that of Berlin.
3z + 4a:-8 = 27. First fill in all the numbers given directly. statement "A and B walk from two towns 27 miles apart until they meet " means the sum of the distances walked by A and B equals 27 miles.e. together.
and Massachusetts has one more than California and Colorado If the three states together have 31 electoral votes.
. then x 2 = number of hours B walks. Since in uniform motion the distance is always the product of
rate
and time.
Find the value of each. or time.
3x
+
4 (x
2)
=
27. such as length. number of miles A
x
x
walks. After how many hours will they meet and how
E. width.
of
arid the value of the iron
was $300.72
of
ELEMENTS OF ALGEBRA
$ 750. i.
= 35. = 5.
has each state
?
If the example contains Arrangement of Problems.
how many
100. and distance.
The copper had twice
the value of the gold. we obtain 3 a. and quantities
area.000. 8 x = 15.g.000. but stops 2 hours on the way. and 4 (x But the 2) for the last column. speed. 3 and 4.
Hence
Simplifying.
start at the same hour from two towns 27 miles walks at the rate of 4 miles per hour. Let x = number of hours A walks.
14.000. and A walks at the rate of 3 miles per hour without stopping. number of hours. it is frequently advantageous to arrange the quantities in a systematic manner.
A and B
apart.
7
Uniting.
Dividing.000 more than that
the copper. of 3 or 4 different kinds.
B
many
miles does
A
walk
?
Explanation.
California has twice as
many
electoral votes as Colorado.
What are the
two sums
5. and how far will each then have traveled ?
9. and the cost
of silk
of the auto-
and 30 yards of cloth cost together much per yard as the cloth. The second is 5 yards longer than the first. but as two of them were unable to pay their share. how much did each cost per yard ?
6.
Find the share of each. together bring $ 78 interest.
as a
4.
Twenty men subscribed equal amounts
of
to raise a certain
money. and in order to raise the required sum each of the remaining men had to pay one dollar more. the area would remain the same.
Find the dimen-
A
certain
sum
invested at 5
%
%. After how many hours will B overtake A.
sum $ 50
larger invested at 4
brings the same interest Find the first sum. twice as large. each of the others had to pay
$ 100 more.
invested at 5 %.
A
sets out later
two hours
B
.
1.
A
If its length
rectangular field is 2 yards longer than it is wide. but four men failed to pay their shares.
If the silk cost three times as
For a part he 7.
mobile.
Ten yards
$
42.55. and the sum Find the length of their areas is equal to 390 square yards. and follows on horseback traveling at the rate of 5 miles per hour.
A sum
?
invested at 4 %.
2. A man bought 6 Ibs. How many pounds of each kind did he buy ?
8. and its width decreased
by 2 yards.
sions of the field.
3.
A
of each.
Six persons bought an automobile. How much did each man subscribe ?
sum
walking at the rate of 3 miles per hour. of coffee for $ 1. paid 24 ^ per pound and for the rest he paid 35 ^ per pound. and a second sum.74
ELEMENTS OF ALGEBRA
EXERCISE
38
rectangular field is 10 yards and another 12 yards wide. were increased by 3 yards.
The
distance from
If a train starts at
.
A
and
B
set out
direction.LINEAR EQUATIONS AND PROBLEMS
v
75
10. and from the same point. but
A has
a start of 2 miles. Albany and travels toward New York at the rate of 30 miles per hour without stopping. and B at the rate of 3 miles per hour.
A
sets out
two hours
later
B
starts
New York to Albany is 142 miles. how must B walk before he overtakes A ?
walking at the rate of 3 miles per hour. and another train starts at the same time from New York traveling at the rate of 41 miles an hour.
walking at the same time in the same If A walks at the rate
of 2
far
miles per hour. After how many hours.will they be 36 miles apart ?
11. how many miles from New York will they meet?
X
12. traveling by coach in the opposite direction at the rate of 6 miles per hour.
expression is rational with respect to a letter. if. if this letter does not occur in any denominator.
An
expression
is integral
and rational with respect
and rational.
which multiplied together
are considered factors. 5.
76
. if it is integral to all letters contained in it. a.
it is
composite. if it does contain
some indicated root of
.
we
shall not. if it contains
no other
factors (except itself
and unity)
otherwise
. at this
6
2
. An expression is integral with respect to a letter.
vV
.
An
after simplifying.
a. but fractional with respect
103.
a-
+
2 ab
+ 4 c2
.CHAPTER
VI
FACTORING
101. consider
105. 6.
The prime
factors of 10 a*b are 2. it contains no indicated root of this letter
.
stage of the work.
irrational.
a2
to 6.
104.
The
factors of
an algebraic expression are the quantities
will give the expression. as.
J Although Va'
In the present chapter only integral and rational expressions
b~
X
V
<2
Ir
a2
b'
2
2
?>
.
this letter.
+ 62
is
integral with respect to a.
a factor of a 2
A
factor is said to be prime.
-f-
db
6
to b.
\-
V&
is
a
rational with respect to
and
irrational with respect
102.
it fol-
lows that every method of multiplication will produce a method
of factoring.
2 4 x + 3) is factored if written (x' would not be factored if written x(x and not a product.
01.g.
77
Factoring
is
into its factors. for this result is a sum.62 + &)(a 2
. .3 6a + 1).9 x2 y 8 + 12
3 xy
-f
by
3
xy\
and the quotient
But.
.
It (a.
x.
The factors
of a
monomial can be obtained by inspection
2
The prime
108.3 sy + 4 y8).)
Ex.
POLYNOMIALS ALL OF WHOSE TERMS CONTAIN A COMMON FACTOR
(
mx + my+ mz~m(x+y + z).
An
the process of separating an expression expression is factored if written in the
form of a product.
Ex.
?/.
it
follows
that a 2
.
Factor
14 a*
W-
21 a 2 6 4 c2
+ 7 a2 6
2
c2
7
a2 6 2 c 2 (2 a 2
. 2.
in the
form
4)
+3.
110.
1.
Hence
6 aty 2
= divisor x quotient.
E.62
can be
&).
factors of 12
&V
is
are 3. 2.
Factor G ofy 2
.
55. 8) (s-1).
y.9 x if + 12 xy\
2
The
greatest factor
common
2
to all terms
flcy*
is
8
2
xy'
.FACTORING
106. dividend
is
2 x2
4
2
1/
.
Divide
6
a% . or that a
=
6)
(a
= a . since (a + 6) (a 2 IP factored.
107.
Since factoring
the inverse of multiplication.
TYPE
I.9 x2^ + 12 sy* = 3 Z2/2 (2 #2 . x.
2.
or
Factoring examples may be checked by multiplication by numerical substitution.
109.
Hence z6 -? oty+12 if= (x -3 y)(x*-4 y ). it is advisable to consider the factors of q first.
We may consider
1.30 = (a . the student should first all terms contain a common monomial factor.11 a + 30.
2. If q is negative. but only in a limited number
of ways as a product of two numbers. + 30 = 20.
but of these only
a:
Hence
2
.1 1 a
tf
a 4.
. or 7 11. a 2 .
79
Factor a2
-4 x .
Factor a2
.1 afy 8 The two numbers whose product is equal to 12 yp and whose sum equals 3 8 7 y are -4 y* and -3 y*.
11 a2 and whose sum The numbers whose product is and a. the two numbers have both the same sign as p.
4.4 .
EXERCISE
Besolve into prime factors :
40
4.
.
+
112.11) (a
+
7).
5.
as p.
Since a number can be represented in an infinite number of ways as the sum of two numbers.11.
3.
77 as the product of 1 77. the two numbers
have opposite
signs.5) (a . of this type.
2
6.11 a
2
.a).
m -5m + 6. or 11 and 7 have a sum equal to 4.
Therefore
Check.G) = .5) (a 6).4 x .
is
The two numbers whose product and -6.
tfa2 -
3. however.
.
Ex. 2 11 a?=(x + 11 a) (a.6 = 20.
or
77
l.
and the greater one has the same sign
Not every trinomial
Ex.
11
7. Factor x? .
If
30 and whose
sum
is
11 are
5
a2
11
a = 1.FACTORING
Ex.
Factor
+ 10 ax .
determine whether
In solving any factoring example.
Ex..77 =
(a. and (a . If q is positive. Hence fc -f 10 ax
is
10 a are 11 a
-
12 /. can be factored.
If
the factors
a combination should give a sum of cross products.1). 64 may be considered the
:
product of the following combinations of numbers 1 x 54.
Ex. X x 18.
The
and
factors of the first term consist of one pair only.17 x
2o?-l
V A
5
-
13 a
combination
the correct one.
The work may be shortened by the
:
follow-
ing considerations
1.
.13 x + 5 = (3 x . none of the binomial factors can contain a monomial factor. exchange the
signs of the second terms of the factors. and r is negative. If py? -\-qx-\-r does not contain any monomial factor. which has the same absolute value as the term qx.5) (2 x . we have to reject every combination of factors of 54 whose first factor contains a 3.
and that they must be negative. or
G
114.
. If p is poxiliw.
the
If p and r are positive. the signs of the second terms are minus. viz. Since the first term of the first factor (3 x) contains a 3.
11 x
2x.FACTORING
If
81
we consider that the
factors of -f 5
as
must have
is
:
like signs.e-5
V A
x-1
3xl \/ /\
is
3
a.
Factor 3 x 2
. 3 x and x. 54 x 1. but the opposite sign.
a.
all
it is not always necessary to write down combinations.
3. the second terms of the factors have same sign as q.
sible
13 x
negative. 2.5 . 9 x 6. 6 x 9. 2 x 27.31 x
Evidently the
last
2
V A
6.83 x
-f-
54. and after a little practice the student possible should be able to find the proper factors of simple trinomials
In actual work
at the first trial. all pos-
combinations are contained in the following
6x-l
x-5 . 18 x 3. then the second terms of
have opposite signs. 27 x 2. Hence only 1 x 54 and 2 x 27 need
be considered.
-
23 3
.
C.
5
s
7
2
5. of 6 sfyz. F. The H.
II
2
. of (a
and (a
+
fc)
(a
4
is
(a
+ 6)
2
.
54
-
32
. F. F.
8
. C.
89
.
15
aW.
3.
25
W. and prefix it as a coefficient to H.
5
7
34 2s
. of
two or more monomials whose factors
. find by arithmetic the greatest common factor of the coefficients. F.
F.
33
2
7
3
22 3 2
.
3
.
121.
2
2
.
+
8
ft)
and
cfiW is
2
a 2 /) 2
ft)
.
Two
common
factor except unity
The H. aW.
The student should note
H. of
:
48
4. C.
13 aty
39 afyV.
are prime can be found by inspection.CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
HIGHEST COMMON FACTOR
120. of a 7 and a e b 7
. is the lowest
that the power of each factor in the power in which that factor occurs in any
of the given expressions.
5
2
3
. C. F. of
aW.
6. F.
C. F.
C. and GO aty 8 is 6 aty.
2. F.
the algebraic factor of highest degree common expressions to these expressions thus a 6 is the II. 12 tfifz. C.
. C.
expressions which have no are prime to one another.
5.
24
s
.
The
highest
is
common
factor (IT.
122. C. The H. F. of a 4 and a 2 b is a2
The H.
EXERCISE
Find the H.) of
two or more
. If the expressions have numerical coefficients. C. Thus the H. of the algebraic expressions.
a^c8
3
.
Hence the L. 60
x^y'
2
. C. M.
127.C.M.
The
L. If the expressions have a numerical coefficient.
=4 a2 62 (a2 . ory is the L.
M. C. L. M.
Find the L.
6
c6 is
C a*b*c*. C. M.6)2.
of 12(a
+
ft)
and (a
+ &)*( -
is
12(a
+ &)( .
The
lowest
common
multiple (L.
etc.
Ex.
2. C. C.
4 a 2 &2
_
Hence.
Common
125.
C.
A
common
remainder. but opposite
.
M.M. each set of expressions has
In example ft). is equal to the highest power in which it occurs in any of the
given expressions.
Ex.
2
The The
L.) of
two or more
expressions is the common multiple of lowest degree. which
also
signs.
&)
2
M. find by arithmetic their least common multiple and prefix it as a coefficient to the L. C.
. C. C.
NOTE.
126. two lowest common multiples. M.(a + &) 2 (a
have the same absolute value. L.C. of as -&2 a2 + 2a&-f b\ and 6-a.
M of the algebraic expressions. resolve each expression into prime factors and apply the method for monomials. To find the L.
M. Obviously the power of each factor in the L.
Find the L.
of 3
aW. thus.
2 multiples of 3 x
and 6 y are 30 xz y.
1.
. of the
general. of tfy and xy*.
300 z 2 y.6 3 ).LOWEST COMMON MULTIPLE
91
LOWEST COMMON MULTIPLE
multiple of two or more expressions is an which can be divided by each of them without a expression
124.
= (a -f
last
2
&)'
is
(a
-
6) . of several expressions which are not completely factored.
of 4 a 2 6 2 and 4 a 4
-4 a 68
2
. M.
128. C.
and
i
x mx = my y
terms
A
1. the value of a fraction is not altered by multiplying or dividing both its numerator and its denominator by the same number.
successively all
2
j/' .
common
6
2
divisors of
numerator and denomina-
and z 8
(or divide the terms
.
Reduce
~-
to its lowest terms. etc. only positive integral numerators shall assume that the
all
arithmetic principles are generally true for
algebraic numbers.
a?.
fraction
is
in its lowest
when
its
numerator
and
its
denominator have no
common
factors.
All operations with fractions in algebra are identical
with the corresponding operations in arithmetic. as 8.
The dividend a is called the numerator and the The numerator and the denominator
are the terms of the fraction. If both terms of a fraction are multiplied or divided by the same number) the value of the fraction is not altered.
Remove
tor.
rni
Thus
132.
TT
Hence
24
2 z = --
3x
. C.
Ex.
a b
= ma
mb
.
the product of two fractions is the product of their numerators divided by the product of their denominators.
130.CHAPTER
VIII
FRACTIONS
REDUCTION OF FRACTIONS
129.
131. Thus.
and denominators are considered.ry ^
by
their H.
an indicated quotient. thus -
is identical
with a
divisor b the denominator.
A
-f-
fraction is
b. but we
In arithmetic. F. however.
D. multiplying the terms of
22
.
we have
-M^.
. and 6rar 3 a? kalr
.
and
Tb reduce fractions to their lowest common denominator.
we have
the quotients (x
1).
mon
T denominator.C.3) (-!)'
=
.
3 a\ and 4
aW
is
12 afo 2 x2 .-1^22
' . and
135.
1.
^
to their lowest
com-
The
L.~16
(a
+ 3) (x.
we may extend this method
to integral expressions. Divide the L.96
134.
by any quantity without altering the value of the fraction.
Multiplying these quotients by the corresponding numerators and writing the results over the common denominator.C.
.
TheL. we may use the same process as in arithmetic for reducing fractions to the lowest
common
denominator.r
2
2
.
Reduce -^-. we have
(a
+ 3) (a -8) (-!)'
NOTE. by the denominator of each fraction.
To reduce
to a fraction with the
denominator 12 a3 6 2 x2 numerator
^lA^L O r 2 a 3
'
and denominator must be multiplied by
Similarly. take the L.
-
by 4
6' .
and the terms of
***.
1).
Ex
-
Reduce
to their lowest
common
denominator.
Since a
(z
-6 + 3)(s-3)O-l)'
6a.
.3)O -
Dividing this by each denominator. of the denominators for the common denominator. =(z
(x
+ 3)(z.
M. C.
and
(a-
8).
-
of
//-*
2
.
+
3).
ELEMENTS OF 'ALGEBRA
Reduction of fractions to equal fractions of lowest common Since the terms of a fraction may be multiplied
denominator.
Ex.by 3 ^
A
2
' .M.
2>
.
multiply each quotient by the corresponding numerator.M. C.
Since -
= a.102
ELEMENTS OF ALGEBRA
MULTIPLICATION OF FRACTIONS
140. Fractions are multiplied by taking the product of tht numerators for the numerator.
integer.
2
a
Ex.
fractions to integral numbers.
F J Simplify
.
we may extend any
e. multiply the
142. (In
order to cancel
common
factors. Common factors in the numerators and the denominators should be canceled before performing the multiplication.
Simplify 1 J
The
expreeaion
=8
6
.g. each
numerator and denomi-
nator has to be factored. and the product of the denominators for the
denominator.
-x
b
c
=
numerator by
To multiply a fraction by an
that integer.
expressed in symbols:
c
a
_ac b'd~bd'
principle proved for
b
141.
!.
or.
2.)
Ex.
Integral or mixed divisors should be expressed in fractional form before dividing.
1. :
a 4-1
a-b
* See page 272.
The
reciprocal of a
number
is
the quotient obtained by
dividing 1
by that number. invert the divisor and multiply it by the dividend. expression by the reciprocal of the fraction.
* x* -f xy 2
by
x*y
+y
x'
2
3
s^jf\ =
x'
2
x*
.
144. x a + b
obtained by inverting
reciprocal of a fraction
is
the fraction. and the principle of division follows
may
be expressed as
145.y3
+
xy*
x*y~ -f y
8
y
-f
3
2/
x3
EXERCISE 56*
Simplify the following expressions
2
x*
'""*'-*'
:
om
2 a2 6 2
r -
3
i_L#_-i-17
ar
J
13 a& 2
5
ft2
'
u2
+a
.
The The
reciprocal of a
is
a
1
-f-
reciprocal of J
is
|
|.104
ELEMENTS OF ALGEBRA
DIVISION OF FRACTIONS
143.
8
multiply
the
Ex. To divide an expression by a fraction.
.
Divide X-n?/
.
The reciprocal of ?
Hence the
:
+*
x
is
1
+ + * = _*_. To divide an expression by a fraction.
A can do a piece of work in 3 days and B in 2 days.
Ex.
and
12
= the number
over.114
35. = the number of minute spaces the minute hand moves
over.
is
36. hence the question would be formulated After how many minutes has the minute hand moved 15 spaces more than the hour hand ?
Let then
x x
= the required number of minutes after 3 o'clock..20
C.
Multiplying by
Dividing.. 2 3
. = 16^.
then
= 2 TT#.180.
Find
R in terms of C and
TT.
C
is
the circumference of a circle whose radius
R.minutes after x=
^
of
3 o'clock.
days by x and the piece of work while in x days they would do
respectively
ff
~ and and hence the sentence written in algebraic symbols ^.
A would do
each day ^ and
B
j.
of minute spaces the
hour hand moves
Therefore x
~ = the number of minute spaces the minute hand moves
more than the hour hand.
PROBLEMS LEADING TO FRACTIONAL AND LITERAL EQUATIONS 152. When between 3 and 4 o'clock are the hands of
a clock together
?
is
At
3 o'clock the hour hand
15 minute spaces ahead of the minute
:
hand.
. Ex.
12.
100
C.
x
Or
Uniting. In how many days can both do it working together ?
If
we denote
then
/-
the required
number
by
1.
~^ = 15
11 x
'
!i^=15. 1.
ELEMENTS OF ALGEBRA
(a) Find a formula expressing degrees of Fahrenheit terms of degrees of centigrade (<7) by solving the equation
(F)
in
(ft)
Express in degrees Fahrenheit 40
If
C.
. 2.
Clearing.
= 100 + 4 x.
= the
x
part of the
work both do
one day.
the rate of the express train. u The accommodation train needs 4 hours more than the express train. 4x = 80. 3. The speed of an express train is $ of the speed of an If the accommodation train needs 4 accommodation train. what is
the rate of the express train
?
180
Therefore.
32
x
= |. then
Ox
j
5
a
Rate Hence the rates can be expressed.
Explanation
:
If
x
is
the rate of the accommodation train. hours more than the express train to travel 180 miles." gives the equation /I)."
:
Let
x -
= the
required
number
of days.
180
Transposing.
Solving.
in
Then
Therefore. the required
number
of days. and the statement.FRACTIONAL AND LITERAL EQUATIONS
A
in symbols the following sentence
115
more symmetrical but very similar equation is obtained by writing ** The work done by A in one day plus the work done by B in one day equals the work done by both in one day.
fx
xx*
=
152
+4
(1)
Hence
=
36
= rate
of express train.
Ex.
But
in
uniform motion Time
=
Distance
.
or 1J.
money and $10. A man lost f of his fortune and $500. a man had How much money had he
at first?
.
by 6.
The sum
10 years hence the son's age will be
of the ages of a father and his son is 50. and J of the greater Find the numbers.
Find two consecutive numbers such that
9. How much money had he at first?
12
left
After spending ^ of his
^ of his money and $15.
is oO.
Find a number whose third and fourth parts added
together
2.
of his present age.
fifth
Two numbers
differ
2. one half of What is the length
of the post ?
10
ter.
its
Find the number whose fourth part exceeds part by 3.
3.
Two numbers
differ
l to s of the smaller.
are the
The sum of two numbers numbers ?
and one
is
^ of the other. to his son. and one half the greater Find the numbers.
make
21. How
did the
much money
man
leave ?
11.
Twenty years ago A's age was |
age. -|
Find their present ages. which was $4000. and of the father's age.
length in the ground.
is
equal
7.
and 9
feet above water.
by 3.
ceeds the smaller by
4.
Find A's
8.
A man left ^ of his property to his wife. to his daughand the remainder.
9
its
A
post
is
a fifth of
its
length in water. and found that he had \ of his original fortune left.
J-
of the greater
increased by ^ of the smaller equals
6.
ex-
What
5.116
ELEMENTS OF ALGEBRA
EXERCISE
60
1.
after
rate of the latter ?
15. ^ at 5%.FRACTIONAL AND LITERAL EQUATIONS
13. what is the
14. Ex.
A has invested capital
at
more
4%. and an ounce of silver -fa of an ounce.
A can
A
can do a piece of work in 2 days. In how many days can both do it working together ? ( 152. 1. If the rate of the express train is -f of the rate of the accommodation train.)
At what time between 7 and 8
o'clock are the
hands of
a clock together ?
17. 3.) (
An express train starts from a certain station two hours an accommodation train.
. A can do a piece of work in 4 clays. and has he invested if
his animal interest therefrom is
19. what is the rate of the express train? 152.)
22. Ex. Ex. An ounce of gold when weighed in water loses -fa of an How many ounce. If the accommodation train needs 1 hour more than the express train to travel 120 miles.
A man
has invested
J-
of his
money
at
the remainder at
6%. and after traveling 150 miles overtakes the accommodation train.
and losing
1-*-
ounces when weighed in water?
do a piece of work in 3 days.
117
The speed
of an accommodation train
is
f of the speed
of an express train. and B in 4 days.
At what time between 7 and
8 o'clock are the hands of
?
a clock in a straight line and opposite
18.
air.
?
In
how many days can both do
working together
23. and B In how many days can both do it working together
in
?
12 days.
at 4J % and P> has invested $ 5000 They both derive the same income from their How much money has each invested ?
20.
How much money
$500?
4%. 2.
152.
At what time between 4 and
(
5 o'clock are the hands of
a clock together?
16. and
it
B in 6 days. ounces of gold and silver are there in a mixed mass weighing
20 ounces in
21.
investments.
ELEMENTS OF ALGEBRA
The
last three questions
and their solutions differ only two given numbers. therefore. m and n. e.
:
In
how many days
if
can
A
and
it
B
working together do a
piece of
work
each alone can do
(a)
(6)
(c)
in the following
number
ofdavs:
(d)
A in 5. Find the numbers if m = 24 30.g.
B in 5.=
m
-f-
n
it
Therefore both working together can do
in
mn
-f-
n
days. A in 6. Then
ft
i.
25.414. and apply the
method of
170. Find three consecutive numbers whose sum equals m. The problem to be solved. and n = 3. A in 4. if
B
in 3 days.
26.= -. B in 12.
. 3. 2.
we
obtain the equation
m m
-.
Find three consecutive numbers whose sum
Find three consecutive numbers whose sum
last
:
The
two examples are
special cases of the following
problem 27.e.
they can both do
in 2 days. .118
153. n x
Solving. B in 16.
. by taking for these numerical values two general algebraic numbers.
To
and
find the numerical answer. B in 30.
is 42. Hence. Answers to numerical questions of this kind may then be found by numerical substitution. Ex. is 57.
make
it
m
6
A can do this work in 6 days Q = 2.009 918. is A can do a piece of work in m days and B in n days.
6
I
3
Solve the following problems
24. it is possible to solve all examples of this type by one example. A in 6. In how
in the numerical values of the
:
many days
If
can both do
we
let
x
= the
it working together ? required number of days.
(d) 1. the second at the apart. (b) 8 and 56 minutes. 3J miles per hour. 3 miles per hour. the area would be increased by 19 square feet.
The
one:
31. Find the side of the square. the
Two men start at the same time from two towns.721. After how many hours do they rate of n miles per hour.
by two pipes in m and n minutes In how many minutes can it be filled by the respectively. 2 miles per hour. d miles the first traveling at the rate of m.
squares
29. 2 miles per hour. After how many hours do they meet.000.
119
Find two consecutive numbers the difference of whose
is 11. the rate of the
first. (a) 20 and 5 minutes.
last three
examples are special cases of the following
The
difference of the squares of
two consecutive numbers
By using the result of this problem. and the second 5 miles per hour.
:
(c)
64 miles.
is (a)
51.FRACTIONAL AND LITERAL EQUATIONS
28. and how many miles does each travel ? Solve the problem if the distance. and how many miles does each travel ?
32.
Two men
start at the
first
miles
apart. (b) 149. respectively (a) 60 miles. (c) 16. (b) 35 miles.
Find two consecutive numbers -the difference of whose
is 21. respectively. if m and n are. 88 one traveling 3 miles per hour.
A cistern can
be
filled
(c)
6 and 3 hours.001.
34.
meet. 5 miles per hour.
same hour from two towns.
is ?n
.
and
the rate of the second are. two pipes together ? Find the numerical answer.
squares
30. If each side of a square were increased by 1 foot.
33.
.
4J-
miles per hour. solve the following ones Find two consecutive numbers the difference of whose squares
:
find the smaller number.
) The ratio of 12 3 equals 4.
b
is
a
Since a ratio
a
fraction.
The
ratio -
is
the inverse of the ratio -. 6 12 = . a ratio
is
not changed
etc.5.
the symbol
being a sign of division.
E.g.
b. etc.
In the ratio a
:
ft. 158. b is the consequent.
Ex.or a *
b
The
ratio is also frequently
(In most European countries this symbol is employed as the usual sign of division.
:
:
155.
Thus the
written a
:
ratio of a
b
is
.
:
A somewhat shorter way
would be to multiply each term by
120
6.
antecedent. the denominator
The
the
157.
.
1.
terms are multiplied or divided by the same number.
is
numerator of any fraction
consequent.
term of a ratio
a
the
is
is
the antecedent.
" a Thus.CHAPTER X
RATIO AND PROPORTION
11ATTO
154.
b. the antecedent. all principles
relating
to
fractions
if its
may
be af)plied to ratios.
The
first
156. instead of writing
6 times as large as
?>.
The
ratio of
first
dividing the
two numbers number by the
and
:
is
the quotient obtained by
second.
Simplify the ratio 21 3|.
A
ratio
is
used to compare the magnitude of two
is
numbers."
we may
write
a
:
b
= 6. the second
term the consequent.
term
is
the fourth proportional to the
:
In the proportion a b = c c?.
:
a-y
.
12. and c.
17.
5 f hours
:
2.
and
c
is
the third proportional to a and
.
AND PROPORTION
ratio 5
5
:
121
first
Transform the
3J so that the
term will
33
:
*~5
~
3
'4*
5
EXERCISE
Find the value of the following
1.
equal
2.
159. b.
Transform the following
unity
15.
two
|
ratios. a and d are the extremes.
18. either mean the mean proportional between the first and the last terms.
1. The last term d is the fourth proportional to a.
4|-:5f
:
5.
terms.
:
ratios so that the antecedents equal
16:64.
61
:
ratios
72:18.
$24: $8.
16 x*y
64 x*y
:
24 48
xif.
4.
proportional between a
and
c.
3:1}.
3.
8^-
hours.
J:l. and the last term the third proportional to the first and second
161.
3:4.
7|:4 T T
4
.
Simplify the following ratios
7. the second
and fourth terms of a proportion are the and third terms are the means. The last
first three.
62:16.
extremes.
10.
b is the
mean
b.
:
is
If the means of a proportion are equal.
In the proportion a b
:
=
b
:
c.
A
proportion
is
a statement expressing the equality of
proportions.
7f:6J.
:
1.
16a2 :24a&.RATIO
Ex.
27 06: 18 a6.
16. b and c the means.
3
8.
11.
9.
= |or:6=c:(Z are
The
first
160.
6.
Hence the number of men required to do some work.
If
(Converse of
nq. of iron weigh .
:
:
directly proportional
may say. or 8 equals the inverse ratio of 4 3.
163. and the time necessary to do it.
ELEMENTS OF ALGEBRA
Quantities of one kind are said to be directly proper
tional to quantities of another kind.
Clearing of fractions. are
: : :
inversely proportional.
The mean proportional
of their product. In any proportion product of the extremes. 3 4.
ccm. 6 ccm.'*
Quantities of one kind are said to be inversely proportional to quantities of another kind. = 30 grams 45 grams.
!-.
2
165.__(163. of a proportion. then 8 men can do it in 3 days. if the ratio of any two of the first kind is equal \o the inverse ratio of the corresponding two of
the other kind.30 grams. then G ccm.
If 6 men can do a piece of work in 4 days. If the product of two numbers is equal to the product of two other numbers^ either pair may be made the means. and the
other pair the extremes.
q~~ n
.
163. i.122
162.)
mn = pq.
briefly. Hence the weight of a mass of iron is proportional to its volume.
pro-
portional. Instead of u
If 4
or 4 ccm. a b
:
bettveen two
numbers
is
equal to
the square root
Let the proportion be
Then Hence
6
=b = ac. and we
divide both
members by
we have
?^~ E. is equal to the ratio of the corresponding two
of the other kind.e.
ad =
be.) b = Vac. " we " NOTE.
164.
t/ie
product of the means
b
is
equal
to the
Let
a
:
=c
:
d. if the ratio of any two of the first kind.
:
c. of iron weigh 45 grams.
ELEMENTS OF ALGEBEA
State the following propositions as proportions : T (7 and T) of equal altitudes are to each. the volume of a
The temperature remaining
body of gas inversely proportional to the pressure.126
54. and
the time necessary for it. and the time. and the
:
total cost. (c) The volume of a body of gas (V) is
circles are to
each
inversely propor-
tional to the pressure (P).
and the area of the smaller
is
8 square inches.
A
line 7^.
What
will be the
volume
if
the pressure
is
12 pounds per square inch ?
. and the area
of the rectangle. The number of men (m) is inversely proportional to the number of days (d) required to do a certain piece of work.
the area of the larger? the same.
1
(6) The circumferences (C and C ) of two other as their radii (R and A").
(b)
The time a
The length
train needs to travel 10 miles.
56.
and the speed
of the train.
A
line 11 inches long
on a certain
22 miles.
the squares of their radii
(e)
55.
what
58.
(c)
of a rectangle of constant width.inches long represents
map corresponds to how many miles ?
The
their radii. State whether the quantities mentioned below are directly or inversely proportional (a) The number of yards of a certain kind of silk. othei
(a) Triangles
as their basis (b
and
b'). under a pressure of 15 pounds per square inch has a volume of
gas
is
A
16 cubic
feet.
(d)
The
areas
(A and
A') of two circles are to each other as
(R and R').
(d)
The sum
of
money producing $60
interest at
5%. (e) The distance traveled by a train moving at a uniform rate.
57.
areas of circles are proportional to the squares of If the radii of two circles are to each other as
circle is
4
:
7.
so that
Find^K7and BO. What is the greatest distance a person can see from an elevation of 5 miles ? From h miles
the
Metropolitan
Tower (700
feet high) ?
feet
high) ?
From Mount
McKinley (20.
as 11
Let
then
:
1.
. it is advisable to represent these unknown numbers by mx and nx.
is
A line AB.RATIO AND PROPORTION
69. When a problem requires the finding of two numbers which are to each other as m n. produced to a point C. = the second number.
:
Ex. 2.
Therefore
7
=
14
= AC. AB = 2 x. 11 x -f 7 x = 108.
11
x
x
7
Ex.000
168. 18 x = 108. 4 inches long.
Hence
or
Therefore
Hence
and
= the first number.
2 x
Or
=
4. 7 x = 42 is the second number.
Divide 108 into two parts which are to each other
7. 11 x = 66 is the first number.
4
'
r
i
1
(AC): (BO) =7: 5.
Let
A
B
AC=1x.
x=2.
127
The number
is
of miles one can see from an elevation of
very nearly the mean proportional between h and the diameter of the earth (8000 miles). x = 6.
Then
Hence
BG = 5 x.
How many
grams of hydrogen are contained in 100
:
grams
10.
A line 24 inches
long
is
divided in the ratio 3
5. find the number of square miles of land and of water.
Gunmetal
tin.
What
are the parts ?
5. Water consists of one part of hydrogen and 8 parts of
If the total surface of the earth
oxygen. and the longest is divided in the ratio of the other two.
consists of 9 parts of copper and one part of ounces of each are there in 22 ounces of gun-
metal ?
Air is a mixture composed mainly of oxygen and nitrowhose volumes are to each other as 21 79.
:
197. 6.
:
4.
Divide 20 in the ratio 1 m.
14. what are
its
parts ?
(For additional examples see page 279.
7. The three sides of a triangle are respectively a. How many ounces of copper and zinc are in 10 ounces of brass ?
6.
12. If c is divided in the ratio of the other two.)
.
The
total area of land is to the total area of
is
water as
7 18.000 square miles. Brass is an alloy consisting of two parts of copper and one part of zinc. 12.
:
Divide 39 in the ratio 1
:
5. 9. cubic feet of oxygen are there in a room whose volume is 4500
:
cubic feet?
8.
How many
7. How many gen. and c inches.000. and 15 inches.
Divide 44 in the ratio 2
Divide 45 in the ratio 3
:
9. How
The
long are the parts ? 15.128
ELEMENTS OF ALGEBRA
EXERCISE
63
1.
m
in the ratio x:
y
%
three sides of a triangle are 11. 2.
:
Divide a in the ratio 3
Divide
:
7.
of water?
Divide 10 in the ratio a
b.
13.
3.
11.
y =
5
/0 \ (2)
of values. which substituted in (2) gives y both equations are to be satisfied by the same Therefore.
the equations have the two values of
y must be equal.
Hence. there is only one solution.-.
Hence
2s -5
o
= 10 _ ^
(4)
= 3.CHAPTER XI
SIMULTANEOUS LINEAR EQUATIONS
169.y=--|. such as
+ = 10. is x = 7.e. etc. expressing a y.-L
x
If
If
= 0. =. the equation is satisfied by an infinite number of sets Such an equation is called indeterminate.
a?
(1)
then
I.
2 y = . x = 1.
if
there
is
different relation
between x and
*
given another equation.
The
root of (4)
if
K
129
. y = 1.
y
(3)
these
unknown numbers can be found.
If
satisfied
degree containing two or more by any number of values of
2oj-3y =
6. if
. From (3) it follows y 10 x and since
by the same values of x and
to be satisfied
y.
However.
An
equation of the
first
unknown numbers can be the unknown quantities. values of x and y.
24.
Solve
-y=6x
6x
-f
Multiply (1) by
2. y
I
171.
By By
Addition or Subtraction.
174.
to
The two methods
I. same relation. Any set of values satisfying 5 x + 6 y = 60 will also satisfy the equation 3 x -f.
Substitution.
21 y
. 3.26. for they cannot be satisfied by any value of x and y. 30 can be reduced to the same form -f 5 y Hence they are not independent.
viz.
ELIMINATION BY ADDITION OR SUBTRACTION
175.
x
-H
2y
satisfied
6 and 7 x 3y = by the values x = I.
are simultaneous equations.
A
system of two simultaneous equations containing two
quantities is solved by combining them so as to obtain
unknown
one equation containing only one
173. the last set inconsistent. y = 2.
unknown
quantity.
6x
.
~ 50.X.3 y = 80. 6 and 4 x y not simultaneous. and 3 x + 3 y =.
= . for they express the x -f y 10.
cannot be reduced to the same form. Independent equations are equations representing different relations between the unknown quantities such equations
.
The process of combining several equations so as make one unknown quantity disappear is called elimination.
172.
Therefore.
4y
.
(3)
(4)
Multiply (2) by
-
Subtract (4) from (3).
ELEMENTS OF ALGEBRA
A
system
of simultaneous equations is
tions that can be satisfied
a group of equa by the same values of the unknown
numbers.130
170. for they are 2 y = 6 are But 2 x 2. 26 y = 60. The first set of equations is also called consistent.
E.
of elimination
most frequently used
II.
Find the number. and if 396 be added to the number. y
31. the number.)
it is advisable to represent a different letter. + z = 2p.
unknown quantity by
every verbal statement as an equation. however.
2
= 6.
2
= 1(1+6).
.
=
l.y
125
(3)
The solution of these equations gives x Hence the required number is 125. Simple examples of this
kind can usually be solved by equations involving only one
unknown
every
quantity.SIMULTANEOUS LINEAR EQUATIONS
143
x
29.
(
99.
. 1.
Let
x
y z
= the
the digit in the hundreds' place. The digit in the tens' place is | of the sum of the other two digits. Problems involving several unknown quantities must contain.
to express
it is difficult
two of the required
digits in
terms
hence we employ 3
letters for the three
unknown
quantities. as many verbal statements as there are unknown quantities.
y
*
z
30.
and Then
100
+
10 y
+z-
the digit in the units' place.
Ex.
Obviously
of the other
. The sum of three digits of a number is 8.
The
three statements of the problem can
now be
readily expressed in
.
+
396
= 521. and to express
In complex examples. the first and the last digits
will be interchanged.
(1)
100s
+ lOy + z + 396 = 100* + 10y + x.
+2+
6
= 8.
= 2 m.
M=i.
Check. 1 digit in the tens place.
x
:
z
=1
:
2.2/
2/
PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS
183. z + x = 2 n.
1
=
2.
symbols:
x
+
y
+z-
8.
# 4. either directly or implied.
ELEMENTS OF ALGE13KA
If both numerator and denominator of a fraction be
. From (3)
Hence xy
Check.
8
= xy + x xy = xy -f 3 x 2 y = 2. who travels 2 miles an hour faster than B.
3
xand y
I
1
(2)
5. and C travel from the same place in the same B starts 2 hours after A and travels one mile per hour faster than A.
3. the fraction
Let and
then y
is
reduced to
nurn orator. 5_
_4_
A. x 3x-4y = 12. = 8.
(3) C4)
=
24 miles.
Find the
fraction. y = 3. starts 2 hours after B and overtakes A at the same How many miles has A then traveled? instant as B.
3+1 5+1
4_2.
By
expressing the two statements in symbols.144
Ex.
Or
(4)-2x(3).
we
obtain.
x 3
= 24.
+
I
2
(1)
and
These equations give x
Check.
Since the three
men
traveled the
same
distance.
(1) (2)
12.
increased by one.
6
x 4
= 24.
xy
a:
2y 4y
2. the distance traveled by A.
direction.
= the
fraction.
2. the fraction is reduced to | and if both numerator and denominator of the reciprocal of the fraction be dimin-
ished by one.
.
4
x
= 24.
x
y
= the = the
x
denominator
.
=
Hence the
fraction
is
f.
Ex. C.
2. B.
the fraction is reduced
fraction.
part of their difference equals
4.}. and the second increased by 2 equals three times the first. A fraction is reduced to J.
The sum
18
is is
and
if
added
of the digits of a number of two figures is 6.
2.
If 4 be
Tf 3 be
is J. and four times the first digit exceeds the second digit by 3. Find the numbers.
7. both terms.
number by
the
first
3. Find the fraction. its value added to the denominator. the value of the fraction is fa.
1. Find the numbers. the last two digits are interchanged.
Five times a certain number exceeds three times another 11. Find the number.
The sum
of the first
sum
of the three digits of a number is 9. if its numerator and its denominator are increased by 1.
6.
5. and the numerator increased by 4. and the fourth 3. to the number the digits will be interchanged.
183. Four times a certain number increased by three times another number equals 33.
Half the sum of two numbers equals 4. and twice the numerator What is the fracincreased by the denominator equals 15. it is reduced to J.SIMULTANEOUS LINEAR EQUATIONS
EXERCISE
70
145
1.
to
L
<>
Find the
If the
numerator and the denominator of a fraction be If 1 be subtracted from increased by 3. If the denominator be doubled. and the two digits exceeds the third digit by 3.)
added to a number of two digits.
added to the numerator of a fraction. If
9 be added to the number. and the second one increased by 5 equals twice
number.
If the
numerator of a fraction be trebled.
the
number
(See Ex. Find the number.
Find the numbers.
?
What
9. the Find the fraction.
. the fraction equals .
If 27 is
10. the digits will be interchanged. fraction is reduced to \-. and
its
denomi-
nator diminished by one.
tion ?
8.
and B's age is \ the sum of A's and C's ages. If the rates of interwere exchanged. and partly at 4 %. the rate of interest?
18. Find their present ages.grams.
13. How 6 %.
and 5 years ago
their ages is 55. Find the weight of one cubic centimeter of gold and one cubic centimeter of silver. partly at 5% and partly at 4%. A man invested $750.
14. Ten years ago the sum of their ages was 90. Find
the rates of interest. and 4 %.
A
sum
of $10. If the sum of
how
old
is
each
now ?
at
invested $ 5000. and The 6 investment brings $ 70 more interest than the 5
%
%
4%
investments together.
Ten years ago A was B was as
as old as
B
is
old as
will be 5 years hence . the annual interest would be $ 195.146
ELEMENTS OF ALGEBRA
11.
.
What was
the
sum and
rates
est
The sums of $1500 and $2000 are invested at different and their annual interest is $ 190.
and
money and
17. now.
partly at
5 %. Three cubic centimeters of gold and two cubic centimeters of silver weigh together 78 grains.000
is
partly invested at
6%. bringing a total yearly interest of $530.
in 8 years to $8500. What was the amount of each investment ?
A man
%
5%. Two cubic centimeters of gold and three cubic centimeters of silver weigh
together 69 J. 5 %. 12. respectively ?
16. the rate of interest ?
What was
the
sum
of
A sum
of
money
at simple interest
amounted
in 2 years
to $090. and the 5% investment brings $15 more interest than the 4 % investment. and in 5 years to $1125.
19. Twice A's age exceeds the sum of B's and C's ages by 30.
much money
is
invested at
A sum
of
money
at simple interest
amounted
in 6 years
to $8000. a part at 6 and the remainder bringing a total yearly interest of $260. What was the amount of each investment ?
15.
A farmer sold a number of horses. then AD = AF.
the length of
NOTE. what are the angles of the triangle ?
22.
A
r
^
A
circle is inscribed in triangle
sides in D.
25. BD = HE. B find angles a. How many did he sell
of each if the total
number
of animals
was 24?
21. and angle BCA = 70.
It takes
A two hours
longer
24 miles.
Find their
rates of walking. and their difference by GO . and F. BC = 7 inches. and angle e angle/.
Find the parts of the
ABC touching the three sides if AB = 9. and CE If AB = G inches. The sum of the 3 angles of a triangle is 180.
An C touch ing the sides in D. ED = BE. c. and e. andCL4 = 8. he would walk it in two hours less than
than
to travel
B
B. but if A would double his pace. what is
that
=
OF. angle c = angle d.
respectively. receiving $ 100 for each horse.
the three sides of a triangle E.
is
the center of the circum-
scribed circle.
.
In the annexed diagram angle a = angle b. The number of sheep was twice the number of horses and cows together.
On
/).
1
NOTE. cows. If angle ABC = GO angle BAG = 50.
points. BE.SIMULTANEOUS LINEAR EQUATIONS
147
20. and F.
24.
23. for $ 740. and sheep. If one angle exceeds the sum of the other two by 20.
E. and $15 for each sheep.
. are taken so
ABC. and GE = CF.
triangle
Tf
AD.
BC=7. and CF?
is
a circle
inscribed in the
7<7. three
AD = AF. $ 50 for each cow. and F '(see diagram). and AC = 5 inches.
is the abscissa. B. Thus the points A. and point the origin.
the ordinate of point P.
two fixed straight lines XX' and YY' meet in at right angles.
The
of
Coordinates. and r or its equal OA is
..
-3).
or its equal
OM.
It'
Location of a point.
first
3).
PN are given. Abscissas measured to the riyht of the origin.
The
abscissa
is
usually denoted by
line XX' is called the jr-axis. then
the position of point is determined if the lengths
of
P
P3f and
185. and
respectively represented
Dare
and
by
(3 7 4).
2). the ordinate by ?/.
(2. jr. PM. (7.
?/. YY' they-axis. and PN _L YY'. hence
The
coordinates lying in opposite directions are negative. and ordinates abore the x-axis are considered positive .
* This chapter
may
be omitted on a
148
reading.
.
(2. and whose ordinate is usually denoted by (X ?/).
lines
PM
the
and P^V are
coordinates
called
point P.
(3. and PJ/_L XX'.CHAPTER
XII*
GRAPHIC REPRESENTATION OF FUNCTIONS AND
EQUATIONS
184.
PN.
is
The point whose abscissa is a.
186.
GRAPHIC REPRESENTATION OF FUNCTIONS
The
is
149
process of locating a point called plotting the point.
and measure
their
distance.
all all
points
points
lie
lie
whose abscissas equal zero ?
whose ordinates equal zero?
y) if y
10.1). (4.
4)
and
(4. (-2. 3).
2.
4).
Draw
the triangle whose vertices are respectively
(-l.
What
are the coordinates of the origin ? If
187.
=3?
is
If a point lies in the avaxis.
Plot the points:
(4.
(-5. (0.(!. 1). -4).
4)
from the
origin ?
7.e. -3).
Where do Where do
Where do
all
points
lie
whose ordinates
tfqual
4?
9. 11.
4. the mutual dependence of the two quantities may be represented
either by a table or
by a diagram.
.
6.
Plot the points
(6.
What
is
the locus of
(a?. -!).4).
1).
6.)
EXERCISE
1. Graphic constructions are greatly facilitated by the use of cross-section paper. (-4. (4. 0).
whose coordinates are given
NOTE.
the quadrilateral whose vertices are respectively
(4.and(l. (0.
What
Draw
is
the distance of the point
(3.
(-3. 0).
2J-). (4. paper ruled with two sets of equidistant and parallel linos intersecting at right angles.
Graphs.
Plot the points
:
(0.2).
(4.
3.
(-1.
0).
71
2).
.
two variable quantities are so related that
changes of the one bring about definite changes of the other.
12. which of its coordinates
known ?
13.3).
8. 0).
-2). -2). 3). i. (See diagram on page 151.
Plot the points: (-4. (-4.
Thus the first table produces 12 points.
representation does not allow the same accuracy of results as a numerical table.
A graphic
and
it
impresses upon the eye
all
the peculiarities of
the changes better and quicker than any numerical compilations.
1. in like manner the average temperatures for every value of the time.
188. or the curved line the temperature.
we meas1
. and the amount of gas subjected to pressures from
pound
The same data. but it indicates in a given space a great many more
facts than a table.
may be found
on Jan. C.
ABCN
y
the so-called graph of
To
15
find
from the diagram the temperature on June
to be 15
. D. 10
.
.
15. however.
from January 1 to December 1. and the corresponding number in the adjacent column the ordinate of a point.. may be represented graphby making each number in one column the abscissa. A.
By representing
of points.
ically
each representing a temperature at a certain date. we obtain an uninterrupted sequence
etc.
ure the ordinate of F.
Thus the average temperature on May
on April 20. B.150
ELEMENTS OF ALGEBRA
tables represent the average temperature
Thus the following
of
New
volumes
1
Y'ork City of a certain
to 8 pounds.
The engineer.
. the rise and fall of wages.
physician.GRAPHIC REPRESENTATION OF FUNCTIONS
151
i55$5St5SS 3{utt|s33<0za3
Graphs are possibly the most widely used devices of applied matheThe scientist uses them to compile the data found from experiments. the
matics. (b) July 15. the merchant. Daily papers represent ecpnoniical facts graphically.
:
72
find approximate answers to the following
Determine the average temperature of New York City on (a) May 1. concise representation of a
number
of numerical data
is
required. the graph
is
applied. (d) November 20. (c) January 15. Whenever a clear. and to deduce general laws therefrom. as the prices and production of commodities.
EXERCISE
From the diagram
questions
1. uses them. etc.
At what date is the average temperature lowest? the lowest average temperature ?
5. from what date to what date would it extend ?
If
.
When
the average temperature below
C.152
2. ?
9.
During what month does the temperature decrease most
rapidly ?
13.. 1?
11
0. 1 to Oct.
When
What
is
the temperature equal to the yearly average of
the average temperature from Sept.
15.
is
10.?
is
is
the average temperature of
New York
6. ?
-
3..
on
1 to
the
average.
During what month does the temperature change least?
14.
Which month
is
is
the coldest of the year?
Which month
the hottest of the year?
16.
How much warmer
1 ?
on the average
is it
on July 1 than
on
May
17.
1 ?
does
the
temperature
increase from
11.
is
ture
we would denote the time during which the temperaabove the yearly average of 11 as the warm season.. At what date is the average temperature highest the highest average temperature?
?
What What
is
4.
ELEMENTS OF ALGEKRA
At what date
(a) G
or dates
is
New York
is
C.
How
much. (c)
the average temperature oi 1 C. (1)
10
C. (freezing
point) ?
7.
From what
date to what date does the temperature
increase (on the average)?
8.
June
July
During what month does the temperature increase most
?
rapidly
12.
During what months
above 18 C. (d) 9 0.
09 yards. Construct a diagram containing the graphs of the mean temperatures of the following three cities (in degrees Fahren-
heit)
:
21.
NOTE.
Draw
.
Hour
Temperature
.
Represent graphically the populations
:
(in
hundred thou-
sands) of the following states
22. in a similar manner as the temperature graph was applied in examples 1-18.
20. From the table on page 150 draw a graph representing the volumes of a certain body of gas under varying pressures.
a temperature chart of a patient.GRAPHIC REPRESENTATION OF FUNCTIONS
18.
19.
153
1?
When is the average temperature the same as on April
Use the graphs of the following examples for the solution of concrete numerical examples.
Draw
a
graph for the
23. transformation of meters into yards. One meter equals 1.
The
initial cost of
cost of manufacturing a certain book consists of the $800 for making the plates. binding.
Show
graphically the cost of the
REPRESENTATION OF FUNCTIONS OF ONE VARIABLE
189.
x increases will change gradually from
13. etc. e. if 1 cubic centimeter of iron weighs 7.
. the daily average expenses for rent..
ELEMENTS OF ALGEBRA
If
C
2
is
the circumference of a circle whose radius
is J2.
2
is
called
x
2 xy
+ 7 is a function of x.
then
C
irJl. if he sells 0..5
grams.50. 9.
28.
190.154
24.
books from
for printing.
x*
x
19. 1 to 1200 copies.g. the value of a of this quantity will change.
29.50.
3. An expression involving one or several letters a function of these letters.)
T
circumferences of
25. if x assumes
successively the
tively
values
1. if each copy sells for $1. x
7 to 9.
+7
If
will
respec-
assume the values 7.) On the same diagram represent the selling price of the books.
to 20 Represent graphically the weight of iron from cubic centimeters. represent his daily gain (or loss). Represent graphically the distances traveled by a train in 3 hours at a rate of 20 miles per hour.
from
R
Represent graphically the = to R = 8 inches. amount to $8. gas.
function
If the value of a quantity changes.
4.inch. and $..
26.
A
10 wheels a day.50 per copy
(Let 100 copies = about \. 3. etc.
to
27. 2 8 y' + 3 y is a function of x and
y.
(Assume ir~
all circles
>2
2
.
2. 2 . 2 x -f 7 gradually from 1 to 2. Represent graphically the cost of butter from 5 pounds if 1 pound cost $.
If
dealer in bicycles gains $2 on every wheel he sells.
to con struct the graph x of x 2 construct a series of -3 points whose abscissas rep2 resent X) and whose ordi1
tions
. 4).
Graph
of a function.
2
(-1. for x=l.
1
the points (-3.
a*.
. be also represented by a graph.
Draw the graph of x2 -f. The values of func192.
is
A
constant
a quantity whose value does not change in the
same discussion. hence
various values of x
The values of a function for the be given in the form of a numerical table.
may. and (3.0).
and join
the
points in order.
To
obtain the values of the functions for the various values of
the
following arrangement
be found convenient
:
.
-J). 4).
values of x2
nates are the corresponding i. (1. (2.
9). 3 50. x a variable.1). however.g. as
1. If a more exact diagram
is
required.
3
(0.
(-
2.
is
supposed to change. 2.
2).
may
.
9).
(1^.2 x
may
4 from x
=
4. while 7 is a constant.
it is
In the example of the preceding article.
155
-A
variable is a quantity
whose value changes in the
same
discussion. E. Thus the table on page 1G4 gives the values of the functions x 2 x3 and Vsr. construct
'.
Ex.
Q-.e.1).GRAPHIC REPRESENTATION OF FUNCTIONS
191. plot points which
lie
between those constructed above.
etc. to
x = 4.
if
/*
4
>
1i >
>
?/
=
193. -1)..
= 0.)
For brevity. as y.
(-3. = 4. (-2. (4.
4J.20).
.
A
Y'
function of the
first
degree is an integral
rational function
involving only
the
power of the variable. the function
is
frequently represented
by a single letter. and joining in order produces
the graph
ABC..
rf
71
. straight line produces the required graph.
It can be
proved that the
graph is a straight
of a function of the first degree
line.
If
If
Locating
ing
by a
3) and (4. j/=-3.
194. the scale unit of the ordinatcs is taken smaller than that of the x..
Ex. y = 6. 2 4 and if y = x -f. Thus 4x + 7.
2.
r
*/
+*
01
.-.4). Thus in the above example.2 x
.
Draw
y
z x
the graph of
= 2x-3. hence two points are sufficient for the construction
of these graphs. and join(0. 5).
7
. 4).. or ax + b -f c are funclirst
tions of the first degree.
(To avoid
very large ordinatcs.156
ELEMENTS OF ALGEBRA
Locating the points(
4. etc.
24 or x =
P and
Q..
C..
1
C. If two variables x and y are directly proportional. Therefore x = 1.e.
Show
any convenient number).. it is evidently possible Thus to find to find graphically the real roots of an equation. i. we have to measure the abscissas of the intersection of the
195. 9 F.
If
two variables x and y are inversely proportional. then
y = .
A body
moving with a uniform
t
velocity of 3 yards per
second moves in
this
seconds a distance d
=3
1.where x
c is
a constant. the abscissas of 3. 32 F. 14 F.158
24.
ELEMENTS OF ALGEBRA
Degrees of the Fahrenheit
(F.) scale are
expressed in
degrees of the Centigrade (C..
that the graph of two variables that are directly proportional is a straight line passing through the origin (assume
for c
27..
From
grade equal to
(c)
the diagram find the number of degrees of centi-1 F.
that
graph with the o>axis. then
cXj
where
c is a constant.24.
.) scale by the formula
(a)
Draw
the graph of
C = f (F-32)
from
to
(b)
4 F F=l.
25.
to Fahrenheit readings
:
Change
10
C.
GRAPHIC SOLUTION OF EQUATIONS INVOLVING ONE
UNKNOWN QUANTITY
Since we can graphically determine the values of x make a function of x equal to zero.
if c
Draw
the locus of this equation
= 12.
y=
formula graphically. what values of x make the function x2 + 2x 4 = (see 192). Represent 26.
(f
.
1)
and
0). locate points
(0.
produces the
7*
required locus. Represent graphically
Solving for
y ='-"JJ y. y = -l.
fc
= 3.
first
degree. Equations of the first degree are called linear equations.
4)
and
(2.1.
Thus
If
in
points without solving the equation for the preceding example:
3x
s
.
i.
X'-2
Locating the points
(2. 2).
Hence.
If the given equation is of the we can usually locate two
y.
NOTE.
T
.
If
x
=
0.
y=
A
and construct
x
(
-
graphically.2 y ~ 2.
Hence
if
if
x
x
-
2. = 0. and join the required graph.
?/.
Hence we may
join (0.
== 2.
if
y
=
is
0.
?/
=4
AB.
4) and
them by
straight line
AB
(3.
199.2. 0).
Graph
of
equations involving two
unknown
quantities.160
ELEMENTS OF ALGEBRA
GRAPHIC SOLUTION OF EQUATIONS INVOLVING TWO UNKNOWN QUANTITIES
198. we can construct the graph or locus of any
Since
we can
=
equation involving two
to the above form.e.
Ex.
represent graphically equations of the form y function of x ( 1D2).
.
3x
_
4
.
y y
2.
Ex. solve for
?/. Draw the locus of 4 x + 3 y = 12.
unknown
quantities.
and joining by a straight
line. that can be reduced
Thus
to represent
x
-
-
-L^-
\
x
=2
-
graphically. because their graphs are straight lines.
1=0.
AB
but only one point
in
AB
also satisfies
(2).
equation
x=
By measuring
3.
202.
AB
y
= . The coordinates of every point of the graph satisfy the given equation.
parallel have only one point of intersection.
To
find the roots of
the system. The roots of two simultaneous equations are represented by the coordinates of the point (or points) at which their
graphs intersect.
By
the
method
of
the preceding article construct the graphs
AB
and
and
CD
of
(1)
(2) respectively.
Graphical solution of a linear system.
203.
3. P.
201. the point of intersection of the coordinate of P.
The
every
coordinates
of
point in satisfy the equation
(1). viz.57. and every set of real values of x and y satisfying the given equation is represented by a point in
the locus.
Since two straight lines which are not coincident nor simultaneous
Ex. we obtain the roots.
Solve graphically the equations
:
(1)
\x-y-\.15.
and CD.
(2)
.GRAPHIC REPRESENTATION OF FUNCTIONS
161
200. linear equations have only one pair of roots.
Solve graphically the
:
fol-
lowing system
= =
25. i.
Inconsistent equations. if x equals
respectively
0. This is clearly shown by the graphs of (1) arid (2).
and
+ 3).g. e.
3. 3.
4. the point
we
obtain
Ex.
Solving (1) for y. (-4. 0. parallel graphs indicate inconsistent equations.
2.
x2
.
1.
V25
5. the graph
of
points
roots.
Using the method of the preceding para.
4. we of the
+
y*
= 25.5. etc. 5. they are inconsistent.
obtain the graph (a circle)
AB C
joining.
.
4. and joining by a
straight line. there are two pairs of By measuring the coordinates of
:
P and Q we find
204.
Locating the
points
(5.
The equations
2
4
= 0. (4.
P
graphs meet in two and $.
4.
5.e.
Measuring the coordinates
of P. 4.
4.. 2.
There can be no point of
and hence no
roots. = 0. 0) and (0.162
ELEMENTS OF ALGEBRA
graph.
3x
2 y = -6.
(1)
(2)
-C.0). which consist of a
pair of parallel lines.0. construct CD the locus of (2)
of intersection.
In general.5. 0. 3).
Locating two points of equation (2).
2 equation x
3). and
.
4.
(1)
(2)
cannot be satisfied by the same values of x and y.
intersection.9.
4. (-2.
y equals
3. 1. AB the locus of (1).
3.y~ Therefore.
-
4. Since the two
-
we obtain DE.
Evolution
it is
is the operation of finding a root of a quan the inverse of involution.
or
-3
for
(usually written
3)
. etc.
\/a
=
x means x n
=
y
?>
a.
\/"^27=-3.
V
\/P
214. which can be simplified no further.
4
4
. for distinction.
27
=y
means
r'
=
27.CHAPTER XIV
EVOLUTION
213.
109
.
tity
. or y
~
3. quantity
may
the
be either 2wsitive
or negative. (_3) = -27.
215.
a)
4
= a4
. and
all
other numbers are.
Every odd root of a quantity has
same sign as
and
2
the
quantity.
1. it is evidently impossible to express an even root of a negative quantity by Such roots are called imaginary the usual system of numbers. and ( v/o* = a.
V9 = +
3.
= x means
= 6-.
numbers.
It follows
from the law of signs
in evolution that
:
Any
even root of a positive.
2.
or x
&4 . Since even powers can never be negative.
Thus
V^I is an imaginary number. called real
numbers.
for (-f 3) 2
(
3)
equal
0. for (+ a) = a \/32 = 2.
a-\-b
is
the root
if
In most cases.2 ac .
+ 6 + 4a&.
2ab
.
8
.
2 2
218.
a2
+ & + c + 2 a& .
14.
multiplied by b must give the last two terms of the
as follows
square.b 2 2 to its square.
.
The
term
a'
first
2
.
15.
2
49a 8 16 a 4
9.>
13.
and
b. a -f. it is not known whether the given
expression is a perfect square. however. and b (2 a -f b).172
7. i.
The work may be arranged
2
:
a 2 + 2 ab
+ W \a + b
.2 ab + b
.2 &c.72 aW + 81 &
4
.e.
the given expression is a perfect square. let us consider the relation of a -f.
second term 2ab by the double of
by dividing the the so-called trial divisor.
2
.
ELEMENTS OF ALGEBEA
4a2 -44a?> + 121V2 4a
s
.
#2
a2
-
16.
10.
term a of the root
is
the square root of the
first
The second term
of the root can be obtained
a.
12. the that 2 ab -f b 2
=
we have then to consider sum of trial divisor 2 a. 11. In order to find a general method for extracting the square root of a polynomial.
mV-14m??2)-f 49.
8 /-. we obtain the next term of the root 3 y 3 which has to be added to 2 the trial divisor.EVOLUTION
Ex.
Arranging according to descending powers of
10 a
4
a. the first term of the answer.
. 8 a 2
-
12 a
+4
a
-f 2. 8 a 2 Second complete divisor.
First trial divisor.
.
Arrange the expression according to descending powers root of 10 x 4 is 4 # 2 the lirst term of the root. 6 a. 1.
*/''
.
and so
forth. We find the first two terms of the root by the method used in Ex.
173
x*
Extract the square root of 1G
16x4
10 x*
__
.
10 a 4
8
a.
of x. 24# 2 y 3 by the trial divisor Dividing the first term of the remainder.
by division we
term of the
root.
double of this term
find the next
is
the
new
trial divisor.
-
24 a
3
+
25 a 2
-
12 a
+4
Square of 4 a First remainder.
Explanation. The process of the preceding article can be extended to polynomials of more than three terms.
is
As
there
is
no remainder.
Ex. Multiply the complete divisor Sx' 3y 3 by Sy 8 and subtract the product from the remainder. As there is no remainder. 8 a 2 .
.24 afy* -f 9 tf.
4 x2
3
?/
8 is
the required square foot.
\
24 a 3
4-f
a2
10 a 2
Second remainder.24 a + 4 -12 a + 25 a8
s
.
First complete divisor.
. and consider Hence the their sum one term.
219. the required root
(4
a'2
8a
+
2}. 2 Subtracting the square of 4x' from the trinomial gives the remainder '24 x'2 + y.
.
The square
. Second trial divisor. By doubling 4x'2 we obtain 8x2 the trial divisor.
Extract the square root of
16 a 4
.
1.
2. 8 a 2
2.
the preceding explanation it follows that the root has two digits. the first of which is 8.
a 2 = 6400.
a
f>2'41 '70
6
c
[700
+ 20 + 4 = 724
2 a
a2 = +6=
41)
00 00
1400
+ 20 = 1420
4
341 76
28400
=
1444
57 76
6776
. the first of which is 4. Hence the root is 80 plus an unknown number. then the number of groups is equal to the number of digits in the square root.
Find the square root of 524. of 10.
1. of 1.
square root of arithmetical numbers can be found to the one used for algebraic
Since the square root of 100 is 10. and the complete divisor
168.
first
.
2. and the square root of the greatest square in
units. and the first remainder is. Thus the square root of 96'04' two digits. the first of which is 9 the square root of 21'06'81 has three digits.000. etc.1344. etc.
Ex. Therefore 6 = 8.000 is 1000.
7744 80 6400
1
+8
160
+ 8 = 168
1344
1344
Since a
2 a
Explanation.000 is 100. beginning at the
and each group contains two digits (except the last. the integral part of the square root of a number less than 100 has one figure.
As
8
x 168
=
1344.EVOLUTION
220.
Find the square root of 7744. the square root of 7744 equals 88. two figures.
the
consists of
group is the first digit in the root.
175
The
by a method very similar
expressions.
Ex. which may contain one or two).000.. of a number between 100 and 10.
The
is
trial divisor
=
160.176.
= 80.
From
A
will
show the
comparison of the algebraical and arithmetical method given below identity of the methods. Hence if we divide the digits of the number into groups. and we may apply the method used in algebraic process.
12.7 to three decimal places.
places.
The groups
of 16724.
Find the square root of
6/.1T6
221.1 are
Ex.688
4
45 2 70
2 25
508
4064
6168 41)600
41344
2256
222.GO'61.
ELEMENTS OF ALGEKRA
In marking
off groups in a number which has decimal begin at the decimal point. annex a cipher.
we must
Thus the groups
1'67'24.
EXERCISE
Extract the square roots of
:
82
.
in .10.0961
are
'. and if the righthand group contains only one digit.70
6. or by transforming the common fraction into a decimal.
Roots of common fractions are extracted either by divid-
ing the root of the numerator by the root of the denominator.
3.
and the first exceeds the second by 405 square yards.
find a in terms of 6
.
Find the numbers.
If
G=m m
g
.
The
two numbers
(See
is
2
:
3.
and they con-
tain together 30G square feet. 25. The sides of two square fields are as 7 2.
2
:
3.
If 22
= ~^-.
r.
solve for v.
Three numbers are to each other as 1 Find the numbers.
:
6.
may
be considered one half of a
rec-
square units. then
Since such a triangle
tangle.)
of their squares
5. and the two other sides respectively
c
2
contains
c
a and b units.
.
Find the side
of each field.
2
.
4.
A
number multiplied by
ratio of
its fifth
part equals 45.
84
is
Find a positive number which
equal to
its
reciprocal
(
144). its area contains
=a
2
-f-
b2
.180
on
__!_:L
ELEMENTS OF ALGEBRA
a.
is
one of
_____
b
The side right angle. If the hypotenuse
whose angles
a
units of length. 24.
26.
108.
If a 2 4.
EXERCISE
1.
A
right triangle is a triangle.
'
4.
If s
= 4 Trr
'
2
.
If 2
-f 2 b*
= 4w
2
-f c
sol ve for
m.
9
&
-{-
c#
a
x
+a
and
c.
solve for d.
and the sum
The
sides of
two square
fields are as
3
:
5.
=
a
2
2
(' 2
solve for solve for
= Trr
.
solve for
r.
2
.
Find
is
the number.b 2 If s
If
=c
.
28.
228.
3.
22
a.
.
2a
-f-
1
23. Find the side of each field. opposite the right angle is called the hypotenuse (c in the diagram).
2.
and their product
:
150.
29.
27. is 5(5.
Method
of completing the
square.
member can be made a complete square by adding 7 x with another term.
.)
13.
2m. Find the unknown sides and the area. passes in t seconds 2 over a space s yt Assuming g 32 feet. x* 7 x=
10. its surface
(Assume
ir
=
2 .
and the two smaller
11.2
7
.
sides.
Two
circles together contain
:
3850 square
feet.
181
The hypotenuse
of a right triangle
:
is
35 inches.
The area $
/S
of a circle
2
.
4.
Find these
10. we have
of
or
m = |.
is
and the other
two
sides are equal.
7r
(Assume
and their
=
2 7
2
.
The hypotenuse
of a right triangle is 2.
.
9.
make x2
Evidently 7 takes the place 7x a complete square
to
to
which corresponds
m
2
.7 x -f 10 = 0.
24.
The following
ex-
ample
illustrates the
method
or
of solving a complete quadratic
equation by completing the square.)
COMPLETE QUADRATIC EQUATIONS
229. the radius of a sphere whose surface equals
If the radius of a sphere is r. (b) 100 feet?
=
.
The area
:
sides are as 3
4.
Solve
Transposing. in how many seconds will a body fall (a) G4 feet.
Find the
sides.
radii are as 3
14. (b) 44 square feet. and the
other two sides are as 3
4. A body falling from a state of rest.QUADRATIC EQUATIONS
7.
. and the third side is 15 inches.
Find the
radii. let us compare x 2
The
left
the perfect square x2
2
mx -f m
to
2
.
-J-
=
12.
the formula
= Trr
whose radius equals r is found by Find the radius of circle whose area S
equals (a) 154 square inches.
8 = 4 wr2 Find 440 square yards. To find this term.
of a right triangle Find these sides. The hypotenuse of a right triangle is to one side as 13:12.
add
(|)
Hence
2
.
8.
.
=8
r/io?.
= 12.
Solving this equation we obtain
by the method of the preceding
2a
The
roots of
substituting the values of a.
Solution
by formula.
=0.
any quadratic equation may be obtained by 6.
o^
or
-}-
3 ax == 4 a9
7 wr
. -\-bx-\. and c in the general answer.184
ELEMENTS OF ALGEBRA
45
46.
231.
2x
3
4.
ao.
49.
x
la
48.
article.
2
Every quadratic equation can be
reduced to the general form.c
= 0.
0.
area
A
a perimeter of 380
rectangular field has an area of 8400 square feet and Find the dimensions of the field.
8.
7.
of their reciprocals is
4.
Problems involving quadratics have
lems of this type have only one solution.3.
56.
2.
88
its reciprocal
A
number increased by three times
equals
6J.
. feet.
Find two numbers whose difference
is 40.
1.
and whose
product
9.
EXERCISE
1. -2.
Twenty-nine times a number exceeds the square of the 190.9.
is
Find two numbers whose product
288.
55.
-4.2.
two numbers is 4. The
11.
G. Find the sides.
2.
1.
and whose sum
is
is 36.0. and the difference Find the numbers.
Divide CO into two parts whose product
is 875.
3.
Find a number which exceeds
its
square by
is
-|. Find the number.
The
difference of
|.
PROBLEMS INVOLVING QUADRATICS
in general two answers.
189
the equations whose roots are
53.1.
-2.
Find the number. and equals 190 square inches.
54.
The sum
of the squares of
two consecutive numbers
85.
57.
:
3.3. -5.
What
are the
numbers
of
?
is
The product
two consecutive numbers
210. and consequently many prob-
235.3.
number by 10.
58.
52.
-2.
3.
5.
Find
the numbers.QUADRATIC EQUATIONS
Form
51.0.
-2. but frequently the conditions of the problem exclude negative or fractional answers.
its
sides of a rectangle differ by 9 inches.
6.
ELEMENTS OF ALGEBRA
The length
1
B
AB of a rectangle. and lost as many per cent Find the cost of the watch.
and gained as many per Find the cost of the horse. exceeds its widtK AD by 119 feet.
c equals 221
Find
AB and AD.
14. he would have received 12 apples less for the same money. What did he pay for
21. had paid $ 20 less for each horse. sold a horse for $144.190
12. Find the rate
of the train.
as the
16. he would have received two horses more for the same money.
ply between the same two ports. he had paid 2 ^ more for each apple.
watch cost
sold a watch for $ 21.
vessel sail ?
How many
miles per hour did the faster
If 20. start together on voyages of 1152 and 720 miles respectively. it would have needed two hours less to travel 120 miles. Two vessels. What did he pay for each
apple ?
A man bought a certain number of horses for $1200.
A man
A man
sold a
as the watch cost dollars. and lost as many per cent Find the cost of the watch.
.
13. If a train had traveled 10 miles an hour faster.
19.
watch for $ 24.
15. one of which sails two miles per hour faster than the other.
of a rectangle is to the length of the recthe area of the figure is 96 square inches.
A man
cent as the horse cost dollars. a distance One steamer travels half a mile faster than the two hours less on the journey.
other.
Two steamers
and
is
of 420 miles.
If he
each horse ?
.
17. A man bought a certain number of apples for $ 2. At what rates do
the steamers travel ?
18. dollars. and the line BD joining
two opposite
vertices (called "diagonal")
feet. and the slower reaches its destination one day
before the other. and Find the sides of the rectangle. ABCD.
The diagonal
:
tangle as 5 4.10.
How many eggs can be bought for $ 1 ?
236.
By formula.
of the area of the basin. the two men can do it in 3 days. how wide is the walk ?
23.
or x
= \/l = 1. Find and CB. In how many days can B do the work ?
=
26.
is
On the prolongation of a line AC. Equations in the quadratic form can be solved by the methods used for quadratics.
237.
1.
(tf.
B
AB
AB
-2
191
grass plot.QUADRATIC EQUATIONS
22. 30 feet long and 20 feet wide.
^-3^ = 7.
and the area of the path
the radius of the basin. A needs 8 days more than B to do a certain piece of work.) 25. a point taken.
A rectangular
A
circular basin is surrounded
is
-
by a path 5
feet wide.
24.
27. and working together. so that the rectangle. The number of eggs which can be bought for $ 1 is equal to the number of cents which 4 eggs cost.
Find
TT r (Area of a circle .
Solve
^-9^ + 8 =
**
0. as
0.
=9
Therefore
x
=
\/8
= 2. contains B 78 square inches.
.
Find the side of an equilateral triangle whose altitude
equals 3 inches. is surrounded by a walk of uniform width.
Ex. constructed with and CB as sides. and the unknown factor of one of these terms is the square of the unknown factor of the
other. If the area of the walk is equal to the area of the plot.I) -4(aj*-l)
2
= 9. 23 inches long.
EQUATIONS IN THE QUADRATIC FORM An equation is said to be in the quadratic form
if it
contains only two unknown terms.
*
III.
must be
*The symbol
smaller than.
the direct consequence of the defiand third are consequences
FRACTIONAL AND NEGATIVE EXPONENTS
243.
no
Fractional and negative exponents. that a
an
= a m+n
.CHAPTER XVI
THE THEORY OF EXPONENTS
242. (a m ) w
. hence.
= a""
<
.
m
IV.a" = a m n
mn .
The
first
of these laws
is
nition of power. ~ a m -f.
Then the law
of involution. The following four fundamental laws for positive integral exponents have been developed in preceding chapters
:
I. very important that all exponents should be governed by the same laws.
>
m therefore. (a ) s=a m = aw bm
a
. 4~ 3 have meaning according to the original definition of power. while the second of the first. however.
244.
we may choose
for such
symbols any definition that
is
con-
venient for other work.
II.
It is.
for all values
1
of
m and n. such as 2*.
a m a" = a m+t1 . and
.
We assume.
provided
w > n. we let these quantities be what they must be if the exponent law of multiplication is generally true."
means "is greater than"
195
similarly
means "is
. instead of giving a formal definition of fractional and negative exponents.
(ab)
.
25.
etc.
23. 4~ .g.
28.
.
30. as.
29. ml.
'&M
A
27.
-
we
find
a?
Hence we
define a* to be the qth root of of.
a\
26.
n 2 a.
laws.
a*.
Assuming these two
8*.
To
find the
meaning
of
a fractional exponent.
= a. at.
0?=-^.196
ELEMENTS OF ALGEBRA
true for positive integral values of n.
we
try to discover the
let the
meaning of
In every case we
unknown quantity
and apply to both members of the equation that operation which makes the negative.
Hence
Or
Therefore
Similarly. (xy$.
Write the following expressions as radicals :
22. fractional. or zero exponent
equal
x.
a?*.
31.
245. since the raising to a positive integral power is only a repeated multiplication. 3*.
24.
m$.
e.
disappear.
Let
x
is
The operation which makes the fractional exponent disappear evidently the raising of both members to the third power.
(bed)*.
^=(a^)
3*
3
. a .
Factors
may
be transferred
from
the
numerator
to
the
denominator of a fraction.g.
by changing the sign of
NOTE.
. or
the exponent.
etc.
each
is
The
fact that a
if
=
we
It loses its singularity
1 sometimes appears peculiar to beginners.
a8 a
2
=
1
1
.
248. in which
obtained from the preceding one by dividing both
members by
a.
vice versa.
Or
a"#
= l.
an x = a.
cr n.
ELEMENTS OF ALGEBRA
To
find the
meaning
of a negative exponent.198
247.
Let
x=
or".2
=
a2
. e. consider the following equations.
a
a
a
= =
a a a
a1
1
a.
Multiplying both members by
a".
1.
If
powers of
a?.
2.
34.
1 Multiply 3 or
+x
5 by 2 x
x. the term which does not contain x may be considered as a term containing #. The
252.
1.
we wish to arrange terms according to descending we have to remember that.
40.2 d
.
6
35.202
ELEMENTS OF ALGEBRA
32.
lix
=
2x-l
=+1
Ex.
Divide
by
^
2a
3 qfo
4.
Arrange in descending powers of
Check.
powers of x arranged are
:
Ex.
V ra
4/
3
-\/m
33.
it
more convenient to multiply dividend and divisor by a factor which makes the divisor rational.
.
44.
Va
-v/a.
ELEMENTS OF ALGEHRA
(3V5-5V3)
S
.
V3 .
(V50-f 3Vl2)-4-V2==
however.
60. a
VS
-f-
a?Vy
= -\/ -
x*y
this
Since surds of different orders can be reduced to surds of
the same order.y.
(3V3-2Vo)(2V3+V5).
53.
Ex.
(5V2+V10)(2V5-1).V5) ( V3 + 2 VS).
52.
Ex.
(3V5-2V3)(2V3-V3).
(2
45.214
42. Monomial surdn of the same order may be divided by multiplying the quotient of the coefficients by the quotient of the
surd factors.
E.
(5V7-2V2)(2VT-7V2).
a fraction.
46. the quotient of the surds
is
If.
is
1
2.
-v/a
-
DIVISION OF RADICALS
267.
47.
51.
48.
49.
43.
268.
all
monomial surds may be divided by
method.
1. metical problems afford the best illustrations.by the usual arithmetical method.g.
.73205
we
simplify
JL-V^l
V3
*>
^>
division
Either quotient equals .
is
Since \/8
12 Vil
=
2 V*2. the rationalizing factor
x
' g
\/2. is illustrated
by
Ex. called rationalizing the
the following examples
:
215
divisor. e.
Hence
in arithmetical
work
it
is
always best to
rationalize the denominators before dividing.
.
/~
}
Ex.
. Evidently.73205..
Divide 4 v^a by
is
rationalizing factor
evidently \/Tb
hence.
+ 4\/5 _ 12v 3 + 4\/5 V8 V8
V2 V2
269.RADICALS
This method. To show that expressions with rational denominators are simpler than those with irrational denominators.
4\/3~a'
36
Ex.
by V7. however. arithTo find. the by 3 is much easier to perform than the division by
1.
Divide 12 V5
+ 4V5 by V. we have
V3
But
if
1.
VTL_Vll '
~~"
\/7_V77
.57735.
we have
to multiply
In order to make the divisor (V?) rational.
3.
The
2.
Divide
VII by v7.
The
difference of
two even powers should always be
considered as a difference of two squares. if n is even.y n is divisible by x -f ?/.
it
follows from the Factoi
xn y n is always divisible by x y.xy +/).
It
y is
not divisible by
287.
ar
+p=
z6
e.
Ex. if n For ( y) n -f y n = 0.
Two
special cases of the preceding propositions are of
viz.
ELEMENTS OF ALGEBRA
positive integer.
286. if w is odd.
We may
6
n 6 either a difference of two squares or a
dif-
* The symbol
means " and so forth to.
x* -f-/
= (x +/)O .
xn -f.
Factor 27 a* -f
27 a 6
8.
By
we obtain the other
factors.
2.
actual division
n.
and have
for
any positive integral value of
If n
is
odd. If n is a Theorem that
1.230
285. xn y n y n y n = 0.
Factor
consider
m
m
6
n9
.
:
importance.
-
y
5
=
(x
-
can readily be seen that #n -f either x + y or x y.
is
odd.
1.
2 8 (3 a )
+8=
+
288.
2."
.g. For substituting y for x.
2
Ex.
customary to represent this result
by the equation ~
The symbol
304.
.
(a:
Then
Simplifying.
great.242
303. it
is
an
Ex.
Interpretation of
QO
The
fraction
if
x
x
inis
infinitely large. the answer is indeterminate.
By making x
any * assigned
zero.
TO^UU"
sufficiently small.
creases.
Let
2.
oo is
= QQ.
(1). without exception.000
a. however
x approaches the value
be-
comes
infinitely large.
The
~~f
fraction .
(1)
= 0. equation. while the
remaining terms do not
cancelj the root is infinity.
+
I)
2
x2
'
-f
2x
+
1
-x(x + 2)= .
Hence such an equation
identity.
i.
x
-f 2.
and becomes infinitely small.
1.
I.
as
+ l.
and
.
the
If in an equation
terms containing
unknown quantity
cancel.i
solving
a problem
the result
or oo indicates that the
all
problem has no solution.
306.
i. or infinitesimal) This result is usually written
:
305.can be
If
It is
made
larger than
number.
ELEMENTS OF ALGEBRA
Interpretation of ?
e.
be the numbers.
= 10.
1.
The
solution
x
=-
indicates that the problem
is indeter-
If all terms of an minate.
.
of the second exceeds the product of the first
Find three consecutive numbers such that the square and third by 1.increases
if
x
de-
x
creases.
is satisfied
by any number.g.
Or.
(1)
is
an
identity.
ToU"
^-100 a. or that x may equal any finite number.x'2 2 x =
1.decreases
X
if
called infinity.e. cancel.
Hence any number will satisfy equation the given problem is indeterminate.e.
Find the side of each square. The sum of the areas of two squares is 208 square feet. Find the edge of each cube.
255 and the sum of
5. and the edge of one exceeds the edge of the other by 2 centimeters.
and the diago(Ex.
is
the breadth
diminished by 20 inches.)
53 yards.
the
The mean proportional between two numbers sum of their squares is 328.
of a right triangle is 73. Find two numbers whose product whose squares is 514.
6. Find the dimensions of the
field. Find these sides.
is
is
17 and the
sum
4.
146 yards.
Find the
sides of the rectangle.
The hypotenuse
is
the other two sides
7. Two cubes together contain 30| cubic inches.
ELEMENTS OF ALGEBRA
The
difference between
is
of their squares
325.
The area of a
nal 41 feet. But if the length is increased by 10 inches and
12.)
The area
of a right triangle is 210 square feet. the area becomes
-f%
of
the original area.
is 6.
Find the other two
sides. Find the edges.
of a rectangular field
feet. and the edge of one.
8. 148 feet of fence are required.
rectangle is 360 square Find the lengths of the sides. Find the numbers.
103. To inclose a rectangular field 1225 square feet in area.
and the sum of
(
228. and the side of one increased by the side of the other e.
9. increased by the edge of the other. p.
10. equals 4 inches.244
3.
13. and
its
The diagonal
is
is
perimeter
11.
and the
hypotenuse
is 37.
and
is
The area of a rectangle remains unaltered if its length increased by 20 inches while its breadth is diminished by 10 inches.
two numbers Find the numbers. 12. The volumes of two cubes differ by 98 cubic centimeters.
.
Find the
sides. 190.quals 20 feet.
14.
) (Area of circle
and
=
1
16. (Surface of sphere
If a
number
of
two
digits be divided
its digits.
the quotient
is 2.SIMULTANEOUS QUADRATIC EQUATIONS
15. Find the number.
.
is
20 inches.
The
radii of
two spheres
is
difference of their surfaces
whose radius = 47T#2.
and
if
the digits will be interchanged. irR *.)
17.
differ by 8 inches. and the equal to the surface of a sphere Find the radii.
by the product of 27 be added to the number. their areas are together equal to the area of a circle whose radius is 37 inches. Find the radii.
245
The sum of the radii of two circles is equal to 47 inches.
to produce the nth term.
309. 3. each term of which. 3 d must be added to a. and d.
series 9. The progression is a.
-f
.7.
added to each term to obtain the next one.
+
2 d.
-4.... The first is an ascending.) is a series..
:
7.
to
A series
is
a succession of numbers formed according
some
fixed law.
of a series are its successive numbers. to produce the 4th term. (n 1) d must be added to a.
Since d
is
a
-f
3
d. P. is derived from the preceding by the addition of a constant number.11 246
(I)
Thus the 12th term of the
3
or 42. the first
term a and
the
common difference d being given. 15 is 9 -f.
..
To
find the
nth term
/
of an A. An arithmetic progression (A. to produce the 3d term... except the first. 12.
of the following series is
3. a
3d. P..
.
to each
term produces the next term. progression.CHAPTER XX
PROGRESSIONS
307. .
a. P.
a
+
d. a -f d. a + 2 d.. a
11.
The terms
ARITHMETIC PROGRESSION
308.
10. 16.
.
The common
Thus each
difference is the
number which added
an A.
17.
Hence
/
= a + (n . 19.
The common differences are respectively 4. 2 d must be added to a.1) d. the second a descending.
11..
5.
if
a = 5.. 19.PROGRESSIONS
310. 21. -4^. -24.
first
2
Write down the
(a)
(6)
(c)
6 terms of an A..
1.
.
Find the 5th term of the
4.3 a = -l.
-|.
the last term
and the common difference d being given.-.. 3.
115. 8. d
.
6.
9.
247
first
To
find the
sum s
19
of the first
n terms of an A.
.
Find the 12th term of the
-4. 5.
= 99.' cZ == ... a = 2.8.
-3. 5. 4...
series
. 3..
Find the nth term of the
series 2. the
term
a.
= -2. P.
= I + 49 = *({ +
..
of the series 10.
2*=(a + Z) + (a + l) + (a + l)
2s = n
*
.. 3. 6. 2J.
2
EXERCISE
1.
Which
(6)
(c)
of the following series are in A. .
Find the 7th term of the Find the 21st term
series
. ?
(a) 1.
Or
Hence
Thus
from
(I)
= (+/). 6.16.
2. 8. -10.
6
we have
Hence
.
. 7.
1.
series 2...
7.
.
8. P. d = 3.
= a + (a
Reversing the order.
9..
Find the 101th term of the
series 1...
3. 2. -7. 5.
1-J. P.-
(a
+ + (a +
l)
l).
. 99) = 2600.-.4.
(d) 1J.. 2
sum
of the first 60
I
(II)
to find the
' '
odd numbers.
5..
Find the 10th term of the
series 17.
Adding.
and for each than for the preceding one.1 -f 3.
13.
15. In most problems relating to A.
8.
to 8 terms.
to 15 terms.
22. 31. How much does he receive (a) in the 21st year (6) during the first 21 years ?
j
311. the other two may be found by the solution of the simultaneous equations
..
. 2J.(#
1
2) -f (x -f 3) H
to
a terms. striking hours only. P. strike
for the first yard.
11. 16.
16. 11.
18.
21.7 -f
to 12 terms.
Q^) How many times
in 12 hours ?
(&fi)
does a clock. hence if any three of them are given.
12.
+ 3.
2.
.
-.
to 20 terms. 15.
1.
7.
'.
3. 7.
4.
to 7 terms. Jive quantities are involved.
.
to 10 terms.
to 16 terms.
.
+ 2-f-3 + 4 H
hlOO. 11.
$1
For boring a well 60 yards deep a contractor receives yard thereafter 10^ more How much does he receive all
together ?
^S5 A bookkeeper accepts a position at a yearly salary of $ 1000.
. 29.
.
15.
1. 1|.
20. 7.
23. and a yearly increase of $ 120.
1J.
.
:
3. 11.
6.
19.
to 20 terms.5
H + i-f
-f-
to 10 terms.248
Find the
10.
\-n.
to 20 terms.
rf.
1+2+3+4H
Find the sum of the
first
n odd numbers. 12.
Sum
the following series
14.
(x +"l) 4.
ELEMENTS OF ALGEBRA
last
term and the sum of the following series :
.
(i)
(ii)
.
>
2-f
2.
33.
17.
6?
9. Find a Given a = 7.
Between 10 and 6
insert 7 arithmetic
means
.
produced. Find d.
3. n.
14.
How much
. n = 13.
a+
and
b
a
b
5. d = 5. ceding one.
.
and
s.
4.
12. n = 16. Find?. Given a = 1.250
ELEMENTS OF ALGEBRA
EXERCISE
116
:
Find the arithmetic means between
1. = 52.
= 16. s = 70. 78.
I
Find
I
in terms of a. of 5 terms
6.
f?
. = 17. Given a = 4. How much did he save the first month?
19.
13. n = 20.
Between 4 and 8
insert 3 terms (arithmetic
is
means)
so
that an A. n = 17. = 83.
10.
has the series 82.
15. Given a = .
16.
8.
I.
17. Given a = |. n = 4. Find w.
A
$300
is
divided
among 6 persons
in such a
way
that each
person receives $ 10 did each receive ?
more than the preceding
one. n
has the series
^
j
.
f
J 1 1
/
.
man saved each month $2 more than in the pre 18.
11.3. Find d. = 1870. = 45. s == 440. Find a and Given s = 44. and all his savings in 5 years amounted to $ 6540.
y and #-f-5y.
7. P. Find n.
a x
-f-
b
and a
b. Find d and Given a = 1700. 74.
T?
^.
How many terms How many terms
Given d = 3.
m
and
n
2. = ^ 3 = 1.
24.
The progression is a.
.
Hence
Thus the 6th term
l
= ar
n~l
. rs =
s
2
-. 36. 36. 36. is
it
(G. 12. P.
of a G. the first
= a + ar -for ar -f ar Multiplying by r.
Therefore
Thus the sum
= ^ZlD.
2 a.
4.
ar8
r. <zr . ar. a?*2 To obtain the nth term a must evidently be multiplied by
.
E.arn ~ l .
NOTE.. except the
multiplying
derived from the preceding one by by a constant number.)
is
a series each term of
which.
A geometric progression
first. the first term a and
the ratios r being given.
is
16(f)
4
. P. 24. To find the sum s of the first n terms term a and the ratio r being given.
(II)
of the
8 =s
first
6 terms of the series 16. or.
g==
it is
convenient to write formula' (II) in
*.
-2.
ratios are respectively 3..
The
314..
s(r
1)
8
= ar"
7*
JL
a. 108. -I..
..PROGRESSIONS
251
GEOMETRIC PROGRESSION
313.
.
(I)
of the series 16.
2
arn
(2)
Subtracting (1) from
(2). +1.
4-
(1)
.
fl
lg[(i)
-l]
==
32(W -
1)
= 332 J.
or 81
315..
.
4.
the following form 8
nf +
q(l-r")
1
r
.g..
If
n
is less
:
than unity. P.
r
n~ l
...
..
and
To
find the
nth term
/ of
a G. called the ratio.
|.
676..
|.
6.5.
volved .
series
Find the llth term of the Find the 7th term of the
ratio is
^.288.
._!=!>.
8.
117
Which
(a)
of the following series are in G. whose and whose second term is 8.
Evidently the total
number
of terms is 5
+ 2.
288. ?
(c)
2. 1.
r^2.. 36. whose and whose common ratio is 4.
first
term
4.
Hence n
=
7..
.
9.
.
And the
required
means are
18. 72.
2
term
3..*. I
= 670.
series 6. whose
.18.. P.
is 16. P.
f..
\
t
series
.252
ELEMENTS OF ALGEBRA
316.
(it.
is 3.
. 144. the other two be found by the solution of the simultaneous equations :
may
(I)
/=<!/-'.18. 20.l.72.
Find the 7th term of the Find the 6th term of the
Find the 9th term of the
^.. 25.
first
5. 18.54. 80. 9.
.
36.
72.
first
term
is
125 and
whose common
.
To
insert 5 geometric
means between 9 and 576.
-fa.
0.
a
=
I.
+-f%9 %
. 3.
Ex.
.
i 288.
. P. f.
l.
Find the 6th term of the
series J.
(d) 5.
series 5.
.
(b) 1.. 9.
..
.
Write down the first 5 terms of a G.
676
t
Substituting in
= r6 = 64.
144.
7. P...-.
..
Hence the
or
series is
0.
or
7.
EXERCISE
1.
10.
4. 4..
. if any three of them are given. 144.
Find the 5th term of a G. hence.
576.5. In most problems relating to G.
-fa. Jive quantities are in. 36.
series
.4.
. P. + 5.6.
Write down the first 6 terms of a G.
188.
179. 12 m. 6 in each row the lowest row has 2 panes of glass in each window more than the middle row.
younger than his Find the age of
the father.
.
The length
is
of a floor exceeds its width
by 2
feet.
176. side were one foot longer. x*
185.
respectively.
186. was three times that of the younger.
Find the number.
+x-
2. z 2
-92.
is
What are their ages ? Two engines are together
more than the
of 80 horse
16 horse power
other.
The age
of the elder of
it
three years ago of each. Four years ago a father was three times as old as his son is now.
A
each
177. How many are there in each window ?
. 10x 2 192.
190. 4 a 2
y-y
-42.
if
each
increased 2 feet.
father.
+
a.
ELEMENTS OF ALGEBRA
A A
number increased by
3.
-ll?/-102. 189.
3 gives the
same
result as the
numbet
multiplied by
Find the number.
. the
sum
of the ages of all three is 51.-36.
180. A house has 3 rows of windows.
power one of the two Find the power of each.
13 a + 3.56.266
173. and the middle row has 4 panes in each window more than the upper row there are in all 168 panes of glass.
187.
2
2
+
a
_ no. 15 m. z 2 + x . aW + llab-2&.
3 gives the
174.
same
result as the
number
diminished by
175.
and | as old as his Find the age of the
Resolve into prime factors
:
184.
A
boy
is
father.
178.
number divided by
3.
A
the
boy
is
as old as his father
and
3 years
sum
of the ages of the three is 57 years. Find the dimensions of the floor. and the father's present age is twice what the son will be 8 years
hence. Find the age
5 years older than his sister
183.
two boys is twice that of the younger.
dimension
182.
train.
181. the ana of the floor will be increased 48 square feet. + 11 ~ 6.
7/
191.
What
is
the distance?
if
square grass plot would contain 73 square feet more Find the side of the plot.
sister
. and 5 h.
An
The two
express train runs 7 miles an hour faster than an ordinary trains run a certain distance in 4 h.
.
(5 I2x
~r
l
a)
.
4x
a
a
2 c
6
Qx
3 x
c
419.278
410.a)(x b
b)
(x
b
~
)
412. In a
if
and
422.
down again
How
person walks up a hill at the rate of 2 miles an hour. How long is each road ?
423. Tn 6 hours
.
2 a
x
c
x
6
-f c
a
+
a
+
a
+
6
-f
walks 2 miles more than B walks in 7 hours more than A walks in 5 hours.(c rt
a)(x
-
b)
=
0. the order of the digits will be inverted.
421.
a
x
a
x
b
b
x
c
b
_a
b
-f
x
414.
a
x
)
~
a
2 b
2
ar
a
IJ a. 411.
(x
. 18 be subtracted from the number.
(x
-f
ELEMENTS OF ALGEBRA
a)(z
-
b)
=
a
2 alb
=
a
(x
-f
b)(x
2
. Find the number. he takes 7 minutes longer than in going.
-f
a
x
-f
x
-f c
1
1
a-b
b
x
415.
-
a)
-2
6 2a.
mx ~
nx
(a
~
mx
nx
c
d
d
c)(:r
lfi:r
a
b)(x
.
A
in 9 hours
B walks
11 miles
number of two digits the first digit is twice the second.
hour.
418 ~j-o.
A man
drives to a certain place at the rate of 8 miles an
Returning by a road 3 miles longer at the rate of 9 miles an hour. and was out 5 hours. far did he walk all together ?
A
. and at the rate of 3^ miles an hour.c) . Find the number of miles an hour that A and B each walk.
420. x
1
a
x
x1
ab
1
1
a
x
a
c
+
b
c
x
a
b
b
~
c
x
b
416
417.
A sum of money at simple interest amounted in 10 months to $2100.
half the
The greatest exceeds the sum of the greatest and
480.
least
The sum
of three
numbers
is
is
21.
to
.
thrice that of his son
and added to the father's.
486.
years.282
ELEMENTS OF ALGEBRA
476.
whose difference
is
4. and in 18 months to $2180. Find the numbers.
483. If 31 years were added to the age of a father it would be also if one year were taken from the son's age
. In a certain proper fraction the difference between the nu merator and the denominator is 12.
. What is that fraction which becomes f when its numerator is doubled and its denominator is increased by 1. had each at first?
B
B
then has
J
as
much
spends } of his money and as A. and the other number least. Find the principal and the rate of
interest. age. Of the ages of two brothers one exceeds half the other by 4 is equal to an eighth of 482.
if
the
sum of
the digits be multiplied by
the digits will be inverted. the Find their ages. How much money
less
484.
by 4.
481.
485. A sum of money at simple interest amounts in 8 months to $260.
latter
would then be twice the
son's
A
and B together have $6000.
477. and becomes when its denominator is doubled and its numerator increased by 4 ?
j|
478. Find their ages.
479. There are two numbers the half of the greater of which exceeds the less by 2. Find the sum and the rate of
interest. fraction becomes equal to |.
If 1
be added to the numerator of a fraction
it
if 1
be added to the denominator
it becomes equal becomes equal to ^.
Find
the number. Find the
fraction. Find two numbers such that twice the greater exceeds the by 30.
A
spends \ of his. and a fifth part of one brother's age that of the other. and 5 times the less exceeds the greater by 3.
487.
A
number
consists of
two
digits
4. also a third of the greater exceeds half the less by 2. and in 20 months to $275. and if each be increased by 5 the Find the fraction. Find the numbers.
they would have met in 2 hours.
. When weighed in water.
it
separately
?
531. In
circle
A ABC.
touches
and
F respectively. A can do a piece of work in 12 days B and C together can do the same piece of work in 4 days A and C can do it in half the time in which B alone can do it.
and CA=7. A vessel can be filled by three pipes.
. A boy is a years old his mother was I years old when he was born. if the number be increased by Find the number.
and third equals \\ the sum third equals \. Find the numbers. M. Tf and run together.
and B together can do a piece of work in 2 days. and one overtakes the other in 6 hours. L. What are their rates of
travel?
. AB=6.REVIEW EXERCISE
285
525. B and C and C and A in 4 days. in 28 minutes. In how many days can each alone do the same work?
526. his father is half as old again as his mother was c years ago. 37 pounds of tin lose 5 pounds. CD.
E
533.
532. if L and Af in 20 minutes.
530. An (escribed) and the prolongations of BA and BC in Find AD. 90. If they had walked toward each other. and BE. N. and 23 pounds of lead lose 2 pounds. BC = 5. How long will B and C take to do
. (a) How many pounds of tin and lead are in a mixture weighing 120 pounds in air.
sum of the reciprocals of of the reciprocals of the first of the reciprocals of the second and
the
sum
528. Two persons start to travel from two stations 24 miles apart. if and L.
527. and losing 14 pounds when weighed in water? (b) How many pounds of tin and lead are in an alloy weighing 220 pounds in air and 201 pounds in water ?
in 3 days. Find the present ages of his father and mother. Tu what time will it be filled if all run
M
N
N
t
together?
529.
AC
in /). it is filled in 35 minutes. the first and second digits will change places. A number of three digits whose first and last digits are the same has 7 for the sum of its digits. Throe numbers are such that the
A
the
first
and second equals
.
How
is
t /
long will
I
take 11
men
2
t' . i. x 8
549. Draw the graph of y 2 and from the diagram determine
:
+
2 x
x*. 2|.
+
3. then / = 3 and write
=
3.
The values of y.
c. FRANCE.
548. GERMANY. 536.
542.286
ELEMENTS OF ALGEBRA
:
534.
-
3 x.e.
543.
550. formation of dollars into marks.
2.
b.
Draw
the graphs of the following functions
:
538.
from x
=
2 to x
= 4.
to
do the work? pendulum. AND BRITISH ISLES
535.10 marks. 2 x
+
5.
d.
of
Draw
a graph for the trans-
The number
in
of
workmen Draw
required to finish a certain piece
the graph
work
D
days
it
is
from
D
1 to
D=
12. 2 541. the time of whose swing a graph for the formula from / =0
537. 3 x
539. The roots of the equation 2 + 2 x x z = 1.
546. x
2
+
x. One dollar equals 4.
-
3 x.
540. z 2
-
x x
-
5.3
Draw
down
the time of swing for a
pendulum
of length
8 feet. the function. if x = f 1. x 2 544.
547. The value of x that produces the greatest value of y.
x*. The values of x if y = 2. The greatest value of the function.
a.
e.
. x*
-
2
x.
-
7. If
to
feet is the length of a
seconds. 2
-
x
-
x2
. x *-x
+
x
+
1.
545. Represent the following table graphically
TABLE OF POPULATION (IN MILLIONS) OF UNITED STATES.
.
Find four consecutive integers whose product is 7920.
in value. and working together they can build it in 18 days. What number exceeds its reciprocal by {$.
The
difference of the cubes of
two consecutive numbers
is
find them. Find the altitude of an equilateral triangle whose side equals a. Find the price of an apple. 727.
What two numbers
are those whose
sum
is
47 and product
A man
bought a certain number of pounds of tea and
10 pounds more of coffee. he
many
312?
he had waited a few days until each share had fallen $6. 721. If a pound of tea cost 30 J* more than a pound of coffee.
716.
723. 725.
a:
713.
sum is a and whose product equals J.l
+
8
-8
+
ft)'
(J)-*
(3|)*
+
(a
+
64-
+ i.
of a rectangle is 221 square feet and its perimeter Find the dimensions of the rectangle.
714
2
*2
'
+
25
4
16
|
25 a2
711.292
709. paying $ 12 for the tea and $9 for the coffee. if 1 more for 30/ would diminish
720. what is the
price of the coffee per
pound ?
:
Find the numerical value of
728. A man bought a certain number of shares in a company for
$375.
.44#2 + 121 = 0.
ELEMENTS OF ALGEBRA
+36 = 0. Find two consecutive numbers whose product equals 600.
217
.25 might have bought five more for the same money.40 a 2* 2 + 9 a 4 = 0. needs 15 days longer to build a wall than B.
___ _ 2* -5 3*2-7
715. 3or
i
-16 .
The area
the price of 100 apples by $1. **-13a: 2
710.
729.
A
equals CO feet. Find two numbers whose 719. How shares did he buy ?
if
726. 16 x* .
722. In how many days can A build the wall?
718. 717. 724. 12
-4*+
-
8. 2n n 2 2 -f-2aar + a -5 = 0.
square inches.
feet.
two numbers Find the numbers. the area of the new rectangle would equal 170 square feet.
and the sum of
their areas 78$. s(y
932.
The sum
of the perimeters of
sum
of the areas of the squares is 16^f feet.
find
the radii of the two circles. A and B run a race round a two-mile course. *(* + #) =24.
935.
931.
is 3.
944.
rate each
man
ran in the
first
heat. and 10 feet broader. Tf there had been 20 less rows.
and the
difference of
936. Find the side of each two
circles is
IT
square.
diagonal
940.
The sum of two numbers Find the numbers.
y(
934.
942.
ELEMENTS OF ALGEBRA
(*+s)(* + y)=10. How many rows are there?
941. The diagonal of a rectangle equals 17 feet.
two squares equals 140 feet. In the second
heat
A
.
two squares is 23 feet. (y + *) = . feet.102. and B diminishes his as arrives at the winning post 2 minutes before B. there would have been 25 more trees in a row. If each side was increased by 2 feet. The difference of two numbers cubes is 513. In the first heat B reaches the winning post 2 minutes before A.
much and A then
Find at what
increases his speed 2 miles per hour.
Assuming
= -y.000 trees. Find the sides of the rectangle. Find the numbers. The
sum
of the circumferences of
44 inches. (y
(* + y)(y +*)= 50.300
930.
is
3
.
the difference of their
The
is
difference of
their cubes
270. 152. = ar(a? -f y + 2) + a)(* + y
933.
943. (3 + *)(ar + y + z) = 96. Find the length and breadth of the first rectangle. z(* + y + 2) = 76.
and the sum of their cubes
is
tangle
certain rectangle contains 300 square feet. y(x + y + 2) = 133.
.
A
is
938. A plantation in rows consists of 10. The perimeter of a rectangle is 92 Find the area of the rectangle.
the
The sum
of the perimeters of
sum
of their areas equals 617 square feet.
34
939.
2240. + z) =108.
937.
is 20. a second rec8 feet shorter. and the Find the sides of the and
its
is
squares. and also contains 300 square feet.
+ z)=18.
is
407 cubic feet. A number consists of three digits whose sum is 14. Find its length and breadth. Find in what time both will do it. set out from two places. at Find the his rate of traveling. triangle is 6.
A and
B. and the other 9 days longer to perform the work than if both worked together. The sum of the contents of two cubic blocks
the
of the heights of the blocks is 11 feet. the digits are reversed.
951.
overtook
miles.
.
unaltered. and that B.REVIEW EXERCISE
301
945. The area of a certain rectangle is 2400 square feet. Find the number. and
its
perim-
948.
. A certain number exceeds the product of its two digits by 52 and exceeds twice the sum of its digits by 53. the area lengths of the sides of the rectangle. and travels in the same direction as A.
sum
Find an edge of
954.
950. A rectangular lawn whose length is 30 yards and breadth 20 yards is surrounded by a path of uniform width. if its length is decreased 10 feet and its breadth increased 10 feet. distance between P and Q. The area of a certain rectangle is equal to the area of a square side is 3 inches longer than one of the sides of the rectangle.
953. When
from
P
A
was found that they had together traveled 80 had passed through Q 4 hours before. at
the same time
A
it
starts
and
B
from
Q
with the design to pass through Q. Find two numbers each of which
is
the square of the other. the square of the middle digit is equal to the product of the extreme digits. If the breadth of the rectangle be decreased by 1 inch and its
is
length increased by 2 inches. Two men can perform a piece of work in a certain time one takes 4 days longer. The square described on the hypotenuse of a right triangle is 180 square inches.
Find the
eter
947.
whose
946.
949.
. the difference in the lengths of the legs of the Find the legs of the triangle. The diagonal of a rectangular is 476 yards. that
B
A
955. Find the width of the path if its area is 216 square yards.
P and
Q. its area will be increased 100 square feet. What is its area?
field is
182 yards. each block.
Two
starts
travelers. was 9 hours' journey distant from P. and if 594 be added to the number.
Find the number.
952.
The sum
982.3 '
Find the 8th
983.
989. P.001
4.1
+
2. Find the
first
term. and the sum of the first nine terms is equal to the square of the sum of the first two.-. Find four numbers in A.
to
n terms.
What
2 a
value must a have so that the
sum
of
+
av/2
+
a
+
V2
+
. 5
11. Insert 8 arithmetic means between
1
and
-. doubling the number for each successive square on the board.
.
992.04
+
.
980.
Find the number of grains which Sessa should have received.-.
"(.
to oo.01
3. Insert 22 arithmetic means between 8 and 54.
990.
Find
n. The
term.)
the last term
the series
a perfect number.
all
A
perfect number
is
a number which equals the sum
divisible. and so on.
first
984.2
.
986.
of n terms of an A. 4 grains on the 3d.001
+
... 2 grains on the 2d.. who rewarded the inventor by promising to place 1 grain of wheat on
Sessa for the
the 1st square of a chess-board.
987. and the
common
difference..
How many
sum
terms of 18
+
17
+
10
+
amount
. P. and of the second and third 03.
to 105?
981.
1.+ lY L V.
v/2
1
+ +
+
1
4
+
+
3>/2
to oo
+
+
. The Arabian Araphad reports that chess was invented by amusement of an Indian rajah. then this
sum multiplied by
(Euclid..
Find four perfect
numbers.
0. Find the value of the infinite product 4
v'i
v7-!
v^5
.
303
979.. such that the product of the and fourth may be 55. Find the sum of the series
988.
of n terms of 7
+
9
+ 11+
is
is
40. is 225.
985.---
:
+
9
-
-
V2
+
.REVIEW EXERCISE
978. named Sheran.
to infinity
may
be 8?
.
If
of
2
of
integers + 2 1 + 2'2
by which
is
it
is
the
sum
of
the series
2 n is prime. The 21st term of an A. P.
are unequal. P.
The sum and sum
. at the same time. c. inches. (I) the sum of the perimeters of
all
squares. The sides of a second equilateral triangle equal the altitudes of the first. P. P. Under the conditions of the preceding example. and so forth to Find (a) the sum of all perimeters.
1000.
AB =
1004. In an equilateral triangle second circle touches the first circle and the sides AB and AC. The other travels 8 miles the first day and After how increases this pace by \ mile a day each succeeding day. ft. many days will the latter overtake the former?
. Each stroke of the piston of an air
air contained in the receiver.
and
if
so forth
What
is
the
sum
of the areas of all circles.
998. areas of all triangles. prove that they cannot be in A. find the series. and G. in this circle a square. 1003. P.
pump removes
J
of the
of air
is
fractions of the original amount contained in the receiver. and so forth to infinity.
1001. in this square a circle. (6) the sum of the infinity. 995. One of them travels uniformly 10 miles a day.
is 4. the sides
of a third triangle equal the altitudes of the second. after how strokes would the density of the air be xJn ^ ^ ne original
density ?
a circle is inscribed.
.
ABC
A A
n
same
sides.
512
996. (a) after 5 strokes. The side of an equilateral triangle equals 2. In a circle whose radius is 1 a square is inscribed.304
ELEMENTS OF ALGEBRA
993. The
fifth
term of a G. P. Insert 3 geometric means between 2 and 162. are
45 and 765
find the
numbers. are 28 and find the numbers. Find (a) the sum of all circumferences. (6) after n
What
strokes?
many
1002. If a. Two travelers start on the same road.
997. The sum and product of three numbers in G.
and the
fifth
term
is
8 times
the second .
third circle touches the second circle and the
to infinity.
994.
of squares of four
numbers
in
G.
999. Insert 4 geometric means between 243 and 32.
very numerous and well graded there is a sufficient number of easy examples of each kind to enable the weakest students to do some work.
xi 4-
373 pages. Ph. so that the Logarithms.
not
The Advanced Algebra is an amplification of the Elementary. than by the
. save Inequalities. but the work in the latter subject
has been so arranged that teachers
who
wish a shorter course
may omit
it
ADVANCED ALGEBRA
By ARTHUR SCHULTZE.
$1.25
lamo.
which have been omitted from the body of the work Indeterminate Equahave been relegated to the Appendix. The more important subjects
tions. proportions and graphical methods are introduced into the first year's course.
i2mo.
$1. and the Summation of Series is here presented in a novel form. etc. Particular care has been bestowed upon those chapters which in the customary courses offer the greatest difficulties to the beginner.D. physics. especially
duction into Problem
Work
is
very
much
Problems and Factoring.
A
examples are taken from geometry.ELEMENTARY ALGEBRA
By ARTHUR SCHULTZE.
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In Factoring. To meet the requirements of the College Entrance Examination Board. comparatively few methods are heretofore. but none of the introduced illustrations is so complex as to require the expenditure of
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Particular care has been the sacrifice of scientific accuracy and thoroughness. there is a sufficient number of easy examples of each kind to enable the weakest students to do some work. bestowed upon those chapters which in the customary courses offer the greatest difficulties to the beginner. The author
grade. The introsimpler and more natural than the
methods given heretofore. great many
A
examples are taken from geometry.ELEMENTARY ALGEBRA
By ARTHUR Sen ULTZE. physics.
not
The Advanced Algebra is an amplification of the Elementary. so that the tions. 64-66
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In Factoring. but none of the introduced illustrations is so complex as to require the expenditure of
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$1. The more important subjects
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-f-
373 pages. proportions and graphical methods are introduced into the first year's course.
HatF leather. especially
duction into Problem
Work
is
very
much
Problems and Factoring. comparatively few methods are
given. Ph.D.
xiv+56a
pages. etc. but the work in the latter subject
has been so arranged that teachers
who
wish a shorter course
may omit
it
ADVANCED ALGEBRA
By ARTHUR SCHULTZE.
has emphasized Graphical Methods more than is usual in text-books of this and the Summation of Series is here presented in a novel form.10
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simple and practical.
izmo. aoo pages. Pains have been taken to give Excellent Figures
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.
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xtt-t
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ositions
7. These are introduced from the beginning 3. Attention is invited to the following important features I.
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. Difficult Propare made somewhat? easier by applying simple Notation . Many proofs are presented in a simpler and manner than in most text-books in Geometry 8. SCHULTZE.
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and successful experience as a teacher
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treated are
:
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.
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.
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.The Teaching
of
Mathematics
in
Secondary Schools
ARTHUR SCHULTZE
Formerly Head of the Department of Mathematics in the High School Commerce. " is to contribute towards book/ he says in the preface. and Assistant Professor of Mathematics in New York University
of
Cloth. New York City.
methods of teaching mathematics the first propositions in geometry the original exercise parallel lines methods of the circle attacking problems impossible constructions applied problems typical parts of algebra.
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pupil's mind the general movements in American history and their relative value in the development of our nation. The book deserves the attention
of history teachers/'
Journal of Pedagogy.
New York
SAN FRANCISCO
BOSTON
CHICAGO
ATLANTA
. All
smaller movements and single events are clearly grouped under these general movements.
i2mo.
diagrams. which have been selected with great care and can be found in the average high school library.
$1. and a full index are provided.AMERICAN HISTORY
For Use
fa
Secondary Schools
By ROSCOE LEWIS ASHLEY
Illustrated.40
is distinguished from a large number of American text-books in that its main theme is the development of history the nation.
but in being fully illustrated with
many excellent maps.
An exhaustive system of marginal references. which put the main stress upon national development rather than upon military campaigns. This book is up-to-date not only in its matter and method.
photographs. Maps.
THE MACMILLAN COMPANY
64-66 Fifth Avenue.
Topics.
Cloth. Studies and Questions at the end of each chapter take the place of the individual teacher's lesson plans. supply the student with plenty of historical
narrative on which to base the general statements and other classifications made in the text.
is
an excellent example of the newer type of
school histories. | 677.169 | 1 |
Lecture 13:
Examples of the DFT and its inverse; DFT of a real valued
signal
Sections 2.2.3, 2.3
In the previous lecture, we considered DFT vectors (or spectra) with a single nonzero
entry. The corresponding time domain-signals are Fourier sinusoids. Thus
Lecture 6:
Aliasing
Sections 1.6
Key Points
Two continuous-time sinusoids having dierent frequencies f and f
(Hz) may, when sampled at the same sampling rate fs, produce sample
sequences having eectively the same frequency. This phenomenon is
known as al
Lecture 10:
Inner Products
Norms and angles
Projection
Sections 2.10.(1-4), 2.12.1
Sections 2.2.3, 2.3
Inner Product
For real-valued vectors, the dot product is also known
as inner product and plays an important role in the
development of vector geometry
Lecture 13:
Introduction to Discrete Fourier Transform
Sections 2.2.3, 2.3
The discrete Fourier transform (DFT) is a powerful computational tool. It allows us to resolve
nite-dimensional signal vectors into sinusoids of dierent frequencies, some of which
Lecture 2:
Lines and circles on the complex plane; complex
multiplication and division
Equations and Curves
|z| - Length of the vector z.
|z1-z2| - Distance between the points z1 and z2.
Geometrical interpretation of the following:
|z z0|= a
a - posi
Lecture 5:
Sampling of Continuous-Time Sinusoids
Sections 1.6
The Sampling formula
Sampling is the recording or grabbing of the
values of a continuous time signal at discrete
time interval.
The sampling formula
x[n]= x(nTs)
x[n] Sequence of samples from
Lecture 1: Complex Numbers
What is a complex number?
A complex number z is a point (or vector) on a
two-dimensional plane, known as the complex
plane and represented by C.
Cartesian Coordinates
The Cartesian coordinates of z are
x = Recfw_z; the real p | 677.169 | 1 |
Maths
GCSE Maths
The new Maths GCSE exam will take place for the first time in the summer of 2017. There is new content at both Higher and Foundation tier in comparison to the old GCSE as well as a greater emphasis on problem solving.
The assessment is in the form of 3 exams all sat at the end of Year 11. They each last 1h30m and two of them allow the use of calculator. Any topic may appear on any of the three papers. There are two tiers of entry – Higher (grades 4-9) and Foundation (grades 1-5).
Please find the programme of study documents below; we currently do the EdExcel GCSE (1MA1). The course is delivered over 3 years and there is regular testing to allow us to track progress and identify areas of weakness that need to be addressed. Detailed analysis is provided after each test, it is crucial that independent study then takes place to work on the areas highlighted.
Support is available in the form of the Maths Clinic which runs each Tuesday between 3 and 4pm in L10 and by using the website detailed in the information sheet below. The CorbettMaths website is also excellent for help videos and practice questions. There is also additional revision for Year 11 every Thursday between 3 and 4pm in L10 (Higher) and L01 (Foundation). | 677.169 | 1 |
Introduction to Advanced Mathematics A Guide to Understanding Proofs
ISBN-10: 0547165382
ISBN-13: 9780547165387 a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed | 677.169 | 1 |
Number: focus on the Numeracy Project and improved coverage of fractions
Geometry: now includes an introduction to angles on parallel lines
Statistics: coverage of the statistical enquiry cycle (PPDAC), and a new emphasis on data distributions, including interpreting dot plots and box plots in context (providing a seamless introduction to the way statistics is covered NCEA).
The chapter on probability includes activities and exercises about informal inference as well as coverage using technology (CAS, spreadsheets and applets) in a probability context.
A Write-on Workbook and an interactive Teacher Guide will be produced to accompany the material. | 677.169 | 1 |
Everything a student needs to succeed in one place. The Student Study Pack contains: * Student Solutions Manual - fully worked solutions to odd numbered exercises. * Pearson tutor Center - Tutors provide one-on-one tutoring for any problem with an answer at the back of the book. Students access the Tutor Center via toll-free phone, fax, or email. Available only to college students in the U.S. and Canada. * CD Lecture Series - A comprehensive set of CD-ROMs, tied to the textbook, containing short video clips of an instructor working key book examples.
"Sinopsis" puede pertenecer a otra edición de este libro.
From the Back Cover:
Exceptionally accessible and user-friendly, this introduction to precalculus features an abundance of interesting real-world applications that relate to readers' everyday lives. Filled with scenarios, examples, study tips, exercises, etc., it takes the intimidation out of learning precalculus, and gets readers up to speed quickly and painlessly. Prerequisites: Fundamental Concepts of Algebra. Equations, Inequalities, and Mathematical Models. Functions and Graphs. Polynomial and Rational Functions. Exponential and Logarithmic Functions. Trigonometric Functions. Analytic Trigonometry. Additional Topics in Trigonometry. Systems of Equations and Inequalities. Matrices and Determinants. Conic Sections. Sequences, Induction, and Probability. For anyone wanting a user-friendly introduction to precalculus.
About the Author:
Bob Blitzer is a native of Manhattan and received a Bachelor of Arts degree with dual majors in mathematics and psychology (minor: English literature) from the City College of New York. His unusual combination of academic interests led him toward a Master of Arts in mathematics from the University of Miami and a doctorate in behavioral sciences fro Nova University. Bob is most energized by teaching mathematics and has taught a variety of mathematics courses at Miami-Dade College for nearly 30 years. He has received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College, and was among the first group of recipients at Miami-Dade College for an endowed chair based on excellence in the classroom. Bob has written Intermediate Algebra for College Students, Introductory Algebra for College Students, Essentials of Intermediate Algebra for College Students, Introductory and Intermediate Algebra for College Students, Essentials of Introductory and Intermediate Algebra for College Students, Algebra for College Students, Thinking Mathematically, College Algebra, Algebra and Trigonometry, and Precalculus, all published by Prentice Hall. | 677.169 | 1 |
A Journey: Student to Teacher & What Lies Beneath
Wednesday, February 15, 2017
Here is the plan that we used for our first cycle. (Most of the activity credits go to Jamie Mitchell and Steph Girvan in the Halton Disctrict School Board - Thank you for sharing your resources - including your blood, sweat and tears guys!)
One of the things we have found as a department is that students often struggle with the algebra portions of this course. Because of this I offered my students an "algebra crash course". I attempted to remind them the core of algebra and solving equations through manipulatives and a clear reminder of what inverse means (i.e. that log is a function, so has an inverse). These should be ingrained ideas that these students have and I find myself often wondering how to best help students at the high school level with these skills. If anyone reading this has any ideas please share!
As you can see in our plan we had two traditional tests in this cycle. We split the algebra portion up into two sections, polynomial & rational functions and logarithmic & trigonometric functions. The last part of the cycle has students explore combinations of functions through investigation of graphs and getting students to do their best to generalize rules for different types of combinations. As a final evaluation in this unit we had student-teacher conferences.
Students had a conference like this one during cycle one as practice (for all of them I was using Google Forms to track and DocAppender to give student immediate access to feedback). For this conference students were given two functions in small groups. They were asked to identify the characteristics of those two functions and then to as a group predict the superposition characteristics of those two functions. On the day of their conferences students rolled a die to get a random second combination. Students were given 5 minutes to prepare and then had 5 minutes to share as much as they could about that combined function. The key was that they were to explain why they believed those were the resulting characteristics, not just to list them.
I found this evaluation very insightful into student reasoning and understanding of characteristics as a whole. It also provided insight into the emphasis that I should consider putting onto the graphical representation of functions in earlier courses. I have started to think that we take for granted what students take away from graphs.
I really enjoyed the experience with conference with these classes and definitely plan to continue working on using them in other courses. Getting students to explain things verbally has an ability to show student learning that reading a written response just cannot do. The power of triangulation of evidence.
Wednesday, February 1, 2017
As mentioned in a post early first semester we made an attempt to spiral the MHF4U0 curriculum at our school. I will try to create some more posts to share more details, so for now this post will just focus on the first cycle we used.
I personally started off with a couple of classes where we did some collaborative problems solving. I wanted to introduce my intention to use visible random grouping (VRG) and vertical non-permanent surfaces (VNPS) in the class. I used this with a couple of fun tasks (such as the Tax Man problem) and then continued with them working on the boards while having them do some review problems together (factoring, radicals). It was a rough beginning. My madness was very new to the students, particularly since I was completely new to the school.
Here is the plan that we used for our first cycle.
(Most of the activity credits go to Alex Overwijk and his team in the Ottawa-Carlton District School Board and to Jamie Mitchell and Steph Girvan in the Halton Disctrict School Board - Thank you for sharing your resources - including your blood, sweat and tears guys!)
The textbook references made above are for the Nelson Advanced Functions book. I very rarely assigned work from the book but students were given the sections as a reference for if they needed it or wanted to do extra practice.
Part way through the cycle (probably about 2/3rds of the way through) I asked the students for some feedback. They were struggling with my use of Desmos Activities and lack of "traditional lectures". We added some more structure to the daily work we were doing. At the start of class we went back to the previous day's lesson (took any questions, which we were already doing) and then co-constructed success criteria based on what they had done. This criteria was added to the lesson plan that the students had access to. I also made a pointed effort to make them read that day's learning goal and asked if anything needed to be clarified. This seemed to help students realize that they were learning.
In retrospect, the vast changes they were going through were a lot. I would create brief google forms for each Desmos Activity the next time to help students consolidate their learning (which would have helped them build their functions portfolio we had asked them to do), essentially they would be exit tickets of some sort. I could collect data for myself while giving students a chance to reflect. And the form could be attached to student documents via DocAppender so that they could have a copy of their own responses.
Our formal evaluation for this cycle was a large group stations task. Students were in groups of 3-4 such that there were 8 groups in one class. There were 8 stations in total (we did 4 per day) that were designed to last approximately 15 minutes each. Of course there turned out to be some they spent more time on than others. Students were to use the time in their groups to work through the problem (i.e. match a graph, table of values, and equation and justify the match) and then record their answer in their own words on their answer sheet.
Students found this to be a very valuable learning tool and, for the most part, the results seemed to align with what we, as teachers, thought that student had shown they knew and could do. They were not big on the fact that it was the only formal evaluation we had done in the first 6 weeks of the course, but appreciated that it was less stressful than a unit test.
In retrospect, the task was too huge for the teachers to deal with all at once. We each had 2 sections x 2 days worth of tasks to go through. It took a lot longer than we anticipated. I would love to do something similar to this again, but would definitely consider splitting it up somehow so that it is not all happening at once. Suggestions are welcome if you have any!
Wednesday, October 12, 2016
This year I am embarking on a new journey - I am working at a different school and have more math in my schedule than I have had since my first year of teaching. As a part of that journey our MHF 4U (Advanced Functions) course team is taking a crack at "spiralling" the course.
Over the summer I spent some time laying out the course to begin to plan. I started with the skeleton Overarching Learning Goals (OLGs) that were created last year for math to come up with OLGs for the course (I wrote about these skeleton OLGs here) so that I would have already wrapped my head around the overall themes of the course.
Then I created a new document to start the actual planning. I pulled the overall expectations (OEs) for the course and the front matter of the math curriculum (math processes (MP)) into the chart by strand and then created a new column where I put in only key words (content & skills) from those OEs and MP. From those words I looked for common themes in the skills/content that I noticed and colour coded them.
Through this process I noticed a major theme in recognizing characteristics of functions and making connections between representations of functions (numerical, graphical, and algebraic). This seemed to be the backbone of a large portion of the course so it made sense to make this into a group of expectations - and cycle 1 was born.
Here are images of that document (they are a work in progress, evolving as we work our way through the course):
Creating the other cycles became largely about noticing the layers involved in the course. I wanted to build the remainder of the course by adding on layers of difficulty, which would allow us to revisit the same concepts. You may have noticed that the second cycle adds on algebraic techniques but is still focused on the same things introduced in cycle 1. This is the purpose of spiralling - students are able to see the same things over multiple exposures to better build their understanding of the material.
Studies are showing that the use of spiralling techniques will help with long-term retention for learners. It is not necessarily about improved results within a particular course, but will help with the foundations moving forward for longer-term success. My hope is that this type of pedagogy can also help with engagement and mindset for learning in the mathematics classroom.
The planning process was somewhat time consuming but was worthwhile for moving forward into the course as I knew what the purpose of the first cycle was and could see the long-term goals. The difficulty was not being able to co-plan with my course team (complicated by it being summer, going into a new school, etc). Now that we are a few weeks into the semester the team is more on the same page and is starting to be able to see the long term plan more easily.
One of the ideas we added to this plan was to have students start and maintain a portfolio where they would put information as it is learned organized by type of function. We are also going to do part of our final 30% as a conference with students - so students have been told that maintaining their portfolio is to aid them with this conference at the end of the semester. The possibilities are exciting.
Wednesday, August 17, 2016
I am currently making my way through Starr Sackstein's Hacking Assessment: 10 Ways to Go Gradeless in a Traditional Grades School. I will likely do a blog when I am finished and will include my takeaways in more detail but one of her lines has inspired me to bring up some provocative conversation.
As are many of you, I am tired of the argument "We have always done it this way." It may be true that there are some thing in our lives that can stay the same year after year and still be the most efficient way to do things but life changes, and often things need to change with it. Most of the stages of my life have come with pretty significant changes and I have also watched the world evolve - computers, internet, cell phones - and would be significantly behind in current knowledge if I had ignored those changes.
Of course, I have also watched the world fail to evolve (lack of change in carbon footprint despite the research and negative effects we are experiencing; decades long wars (in so many cases started by some misunderstanding and a failure to learn from history) that are sometimes ignored by the rest of the world; etc.
I can no longer watch education become something that does not change.
On page 28 Sackstein had me out loud proclaiming "Yes! This is the articulation I have been looking for!" when she said:
In the industrial era, schools were intended to train good
workers, so students went to schools that prepared them to enter the
work force. This model of education valued obedience, conformity,
and rote learning.
We are no longer in the industrial evolution.
Of course, I will not pretend to believe that we do not need workers who can conform and follow specific steps to complete a task, but the majority of work that we need to prepare students for requires people who can be creative, who can think for themselves, and who can solve problems. This is the world I want to prepare my students for - I want them to find success in whatever passion or skill they find for themselves through the use of transferable, valuable skills (not rote learning they can look up on YouTube).
Hopefully my journey can help to bring along more teachers, students, parents, admin, and community members who want to see a change. | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
548 KB|37 pages
Product Description
The following lesson is crafted around "The Special Types of Linear Systems!"
* Students will find this lesson to be the "kingpin" of all Linear System lessons, as it includes quick and efficient ways for one to recognize a Linear System with ONE solution, NO solution, and INFINITELY MANY solutions (please note the cut pieces of lumber on the floor of the carpenter's shop denoting one solution, infinitely many solutions, and no solution).
* Students will surely find themselves "hinging" onto the information.
* There is a real life problem that involves a carpenter who orders materials for two upcoming jobs where students' knowledge learned in this lesson about Linear Systems is required in order to solve.
* A quick three question Follow up assessment is provided so that they may assess their progress, and answers are provided so students receive immediate feedback.
* The lesson is aligned with Common Core expectations, while the slides are clearly outlined for the students' maximum understanding.
* Additionally, a section of GUIDED NOTES is included to encourage active participation in the presentation, and later these notes can be used in preparation for quizzes and tests. | 677.169 | 1 |
Mathematics
Junior Cycle - Mathematics
Subject Group: Science
These subjects demonstrate how to explore nature using carefully planned methods, and teach the basic methods and findings of scientific investigation.
Brief Description:
In Mathematics, you will develop your problem-solving skills and your ability to present logical arguments. You will be better able to use what you learned in Mathematics in real life situations in everyday life and work.
How will Mathematics be useful to me?
Studying mathematics prepares you for business calculations, for handling your money sensibly and for courses in sciences, engineering and technology. You should see mathematics as an opportunity to strengthen your thinking skills.
Note: The Revised Junior Certificate Mathematics Syllabus for Examination in 2015 and in 2016 are available through the 'Course Outline' link below. | 677.169 | 1 |
...The topics learned in middle school prealgebra form a foundation of math skills that are used in every math course that comes later in high school and college curricula. This means that if you let your student continue to struggle with fraction arithmetic, arithmetic using negative numbers, fact... | 677.169 | 1 |
Setting up a Mathematics and/or Statistics Support Centre
The purpose of this booklet is to act as a guide for those interested in setting up or enhancing a mathematics and/or statistics support provision for students (not necessarily students of mathematics or statistics) in addition to their regular programme.
In this guide, a summary of the arguments about why mathematics support is necessary is presented. A series of case studies is presented which show how mathematics support operates in a range of institutions and at a variety of scales.
The second edition of a guide for those interested in establishing and developing Mathematics Support Centres in institutes of higher education. The guide distils the findings of the LTSN funded teaching and learning project Evaluating and Enhancing the Effectiveness of Mathematics Support Centres. | 677.169 | 1 |
Study Guide for Test 1 MATH 2414 (Graves) Test 1 covers Chapter 7 (all sections). 1. Know how to evaluate both defnite and indefnite integrals exactly using the techniques we have discussed. Section 7.5 Exercises, pp. 488–489: 1–80 Chapter 7 Review Exercises, pp. 518–519: 1–40 (IF these are too many problems For you to work, work just the odd numbered ones. ±or extra practice, work the even numbered ones as well.) 2. Know how to use a table oF integrals to evaluate integrals.
This is the end of the preview. Sign up
to
access the rest of the document. | 677.169 | 1 |
Solutions Made Simple
Active Learning
MS Courses
The School of Mathematics & Science offers modules in mathematics, computing and science (physics) for students enrolled in the engineering, technology and business courses. These modules form an important part of the full-time curriculum.
A sound foundation in mathematics, science and computing together with proficiency in their applications, help students develop analytical, logical thinking, and problem solving skills. These provide an important foundation for students to achieve life-long learning in the future.
We also offer courses such as the Specialist Diploma course in Statistics & Data Mining and other short courses to our graduates and the public.
Polytechnic Foundation Programme (PFP)
This is a one-year practice-oriented curriculum offered to polytechnic-bound Normal (Academic) students for a seamless integration into the course of their choice in SP.
Diploma-Plus Programme
MS offers a Certificate in Software Programming & Applications (PCSPA) course under the SP Diploma-Plus Programme. This course aims to provide engineering and business students with higher level skills in computer programming and software applications.
Preparatory and Bridging Programmes
Preparatory Mathematics Programme: This is designed for newly enrolled students with Elementary Mathematics only in their GCE O-level examination. Its aim is to prepare students for their course of study by reviewing some of the fundamental mathematical concepts and skills. Click here for further information.
Bridging Mathematics for ITE Upgraders: This provides ITE upgraders who enrol into SP's engineering courses a structured bridging mathematics programme to strengthen their mathematics foundation to enable them to better cope with the demands of their respective Engineering course. Click here for further information.
Advanced Programmes
MS also offers advanced programmes in Mathematics and Physics for Engineering and Info-Technology students. The training is to help the students build a solid foundation in mathematics to prepare them for further studies. The training will also develop their analytical, logical thinking and problem solving skills.
Click here for further information on the advanced programmes offered to non-engineering schools (DMIT & SB).
Post-Diploma Programme
The Specialist Diploma in Statistics and Data Mining (NSSDM) is a part-time course offered by MS. This course is designed to meet the growing needs for expertise in statistical analysis and data mining using relevant analytical tools and systems.
Short Courses
Preparatory Mathematics for University (PMU): This course aims to strengthen the foundation of polytechnic graduates in mathematics to prepare them for university education, particularly in the field of engineering and IT. It is also designed to bridge the gap between the calculus at the polytechnic diploma level and the calculus at the first year university level. Click here for further information.
Preparatory Physics for University (PPU): This course aims to strengthen the physics foundation of polytechnic graduates to prepare them for university education. Participants will receive the training needed to solve physics problems, building on their knowledge of mathematics and basic calculus. Click here for further information.
A sound foundation in mathematics, science and computing together with proficiency in their applications is indispensable for engineering and business studies. | 677.169 | 1 |
Document
Info
Targeted Math | Probability
This lesson plan is one of 12 that was created for teachers to use with their students who need to brush up on their math skills in order to pass the 2002 GED® Math Test. To see all 12 lesson plans, go to Targeted Math Instruction for the 2002 GED® Test.
Probability is the study of chance, or the likelihood that an event will occur. Probability can't tell us what will happen, only what is likely to happen. Here we examine how we find probability mathematically.
Vocabulary for Statistics and Probability is found in the GED Connection Mathematics workbook, "Program 37," p. 230.
This lesson should take approximately 1 hour and 15 minutes to complete, if all components are utilized. | 677.169 | 1 |
Description - Mathematics for Economists by E. Roy Weintraub
The responses to questions such as 'What is the explanation for changes in the unemployment rate?' frequently involve the presentation of a mathematical relationship, a function that relates one set of variables to another set of variables. It should become apparent that as one's understanding of functions, relationships, and variables becomes richer and more detailed, one's ability to provide explanations for economic phenomena becomes stronger and more sophisticated. The author believes that a student's intuition should be involved in the study of mathematical techniques in economics and that this intuition develops not so much from solving problems as from visualizing them. Thus the author avoids the definition-theorem-proof style in favor of a structure that encourages the student's geometric intuition of the mathematical results. The presentation of real numbers and functions emphasizes the notion of linearity. Consequently, linear algebra and matrix analysis are integrated into the presentation of the calculus of functions of several variables. The book concludes with a chapter on classical programming, and one on nonlinear and linear programming.
This textbook will be of particular interest and value to graduate and senior undergraduate students of economics, because each major mathematical idea is related to an example of its use in economics. | 677.169 | 1 |
Group theory is the branch of mathematics that studies symmetry, found in crystals, art, architecture, music and many other contexts. But its beauty is lost on students when it is taught in a technical style that is difficult to understand. Visual Group Theory assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective.
The more than 300 illustrations in Visual Group Theory bring groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory.
This text approaches the learning of group theory visually. It allows the student to see groups, experiment with groups and understand their significance. It is ideal as a supplement for a first course in group theory or alternatively as recreational reading.
About the Author:
Nathan Carter earned his Ph.D. in mathematics at Indiana University in July 2004. He received the University of Scranton Excellence in Mathematics Award in 1999, an Indiana University Rothrock Teaching Award in 2003, and a Bentley College Innovation in Teaching Award in 2007. Visual Group Theory is his first book, based on lessons learned while writing the software Group Explorer. Like several of his research projects, it puts computers to work to improve mathematical understanding and education. | 677.169 | 1 |
ISBN 13: 9780007170937
GCSE Maths (Do Brilliantly at...)
High GCSE grades are gained through a combination of good knowledge, good understanding and good exam technique. The "Do Brilliantly At..." series is all about exam technique and because it's written by the people who mark the exams, it really should help students improve their performance. The content of this text has been thoroughly updated to match the latest GCSE maths exams and provides examples of exam questions across all the exam boards. With key skills highlighted, lots of questions to try and tips from top examiners, it should give students the confidence they need to do well.
"synopsis" may belong to another edition of this title.
About the Author:
Paul Metcalf if a freelance Consultant in Mathematics and is Principal Moderator for a major examining group. He was formerly a Head of Mathematics and a Deputy Headteacher.585533518
Book Description Collins 19170937
Book Description Collins 19170937
Book Description Collins, 2004. Paperback. Book Condition: Good. Do Brilliantly At - GCSE Maths170937 | 677.169 | 1 |
ISBN-10: 0471359432
ISBN-13: 9780471359432 international experts in the field of integer programming, this text and reference presents the mathematical foundations, theory, and algorithms of discrete optimisation | 677.169 | 1 |
Mathematics for Elementary Teachers II
MTH 116W
Mathematics for Elementary Teachers II
Course Description
Prerequisite: MTH 115 or MTH 110 with a grade of "C" or better. Investigates problem
solving, statistical charts and graphs, geometric figures and properties, and measurement
systems including metric. Reviews fractions, decimals, percents, real numbers, their
operations and properties. Reviews algebra of lines and equations. Includes a variety
of learning styles using manipulatives, calculators and computer application. The
National Council of Teachers of Mathematics Standards are incorporated. (45-0)
Outcomes and Objectives
Solve problems using various problem solving strategies.
Objectives:
Define, illustrate, and utilize various problem solving strategies.
Investigate elementary logic.
Objectives:
Examine and utilize truth tables and deductive (direct) reasoning.
Students will develop their skills in elementary probability.
Objectives:
State and apply the definitions of experiment, event, outcome, and sample space.
Explain what is meant by the probability of an event and distinguish between experimental
and theoretical probability.
Compute probabilities for events with equally likely outcomes.
Use a tree diagram to represent the outcomes in a sample space.
Draw Pascal's triangle and be able to apply it in a binomial experiment.
Students will develop their skills in geometry.
Objectives:
Give analytical descriptions of various types of triangles.
List several properties common to various types of triangles and quadrilaterals.
Illustrate reflection and rotation symmetry of polygons, regular n-gons, and other
shapes.
Define and identify properties of points, rays, lines, planes, line segments and angles.
Describe a) the interior of an angle; b) adjacent angles and ; c) how to measure an
angle with a protractor.
Explain and use the corresponding angles property, the alternate interior angles property
and derive the angle sum in a triangle property.
Determine the measures of central angles, vertex angles, and exterior angles in regular
polygons and discuss the relationships among their measures | 677.169 | 1 |
Friday, November 11, 2011
Reflect (express yourself, your feelings/emotions, give your opinion) on what happened in class today: new assigned seats, working with the clickers, working on your own, kind of practice, lenght of practice, knowing your grade immediately, the grade you got, etc. Please use at least two sentences. Have a wonderful weekend!
This is The Scribe List. Every possible scribe in our class is listed here. This list will be updated every day. If you see someone's name crossed off on this list then you CANNOT choose them as the scribe for the next class.
This post can be quickly accessed from the [Links] list over there on the right hand sidebar. Check here before you choose a scribe for tomorrow's class when it is your turn to do so.
IMPORTANT: Make sure you label all your Scribe Posts properly or they will not be counted.
Monday, October 31, 2011
Hello!!! Welcome to Our Blog!!! This is our space, the place to talk about what's happening in class; to ask a question you didn't get a chance to ask in class; to get copies of a handout you didn't get in class (the course syllabus is below); for parents to find out "How Was School Today;" to share your knowledge with other students. Most importantly it's a place to reflect on what we're learning.
A big part of Learning and Remembering involves working with and discussing new ideas with other people -- THIS is the place to do just that. Use the comment feature below each post, or make your own post, contribute to the conversation and lets get down to some serious blogging!
Algebra 2 is a program to teach the mathematical concepts and methods that the students need to know in order to meet high curriculum standards and succeed on states required tests.We will explore number systems and algebraic operations that involve real and complex numbers.Perform applications for data analysis, statistics, probability, functions and relations.
COURSE CONTENT
The Oklahoma State Department of Education has specific PASS (Priority Academic Student Skills) Objectives for Algebra 2.These PASS Objectives will be tested by the EOI (End of Instruction Test) in the Spring 2012.The objectives are listed below by quarter.
5. Use measures of central tendency & use appropriate measures of variability to analyze data.
6. Use characteristic of Guassian normal distribution and identify how outliners affect representation of data.
COURSE MATERIALS
The school provides calculators, tools and textbook.The student is responsible for taking good care of the tools we use in the classroom.Students should have a notebook (or binder with papers) and pencils.
COURSE POLICIES
1. Absences/Makeup work:When you return from an absence, you are responsible for the following:
Turning in any homework that was due the day(s) of your absence(s).
Reading the textbook section or notes taken or other material that was used as a resource during your absence(s).
According to school and district policy, work, including tests, must be made up within five (5) school days of the absence.Failure to make up work with the regular teacher within the allotted five (5) days will result in the students receiving "NG" (No Grade).
2. Special projects: We will be using projects to explore extended problems that are relevant to us and have real-world connections.For every project I assign, I will provide a scoring rubric that identifies and explains the important components of each project.
3. Classroom rules/expectations:
Be Polite: be respectful of teacher and classmates.
Be Prompt: be in your seat and ready to work when the bell rings
Be Productive: use your time efficiently from bell-to-bell and put forth 100% effort
Be Prepared: arrive with all classroom materials and assignments.
Participate: engage in all classroom activities with a positive attitude.
Others:
Dress code set forth by US Grant High School
Bathroom pass – once a month with school ID
Don't cheat
If a student violates the class rules a warning (or conference) will be given, student will have detention, parents will be called and/or the student will receive an office referral.
No food/drinks/candy
No cell phone/electronic devices
Ask questions
No sleeping
4. Grading Policy
Grade distribution: Quarter grades will be calculated as follows:
Tests 43%
Quiz 15%
Final 12%
Practice work (daily, homework) 20%
Projects 10%
The letter grading system is as follows:
100-90% - A
89-80% - B
79-70% - C
69-60% - D
59% and below - F
5. Extra help
Get extra help when you need it.I will be available for tutoring after school. I will be happy to arrange it for anyone who requests it or when assigned.
====================================================================
Please cut and return this part to your teacher:
Algebra 2 – TB _______Date:____________________
I certify that I received and read the Algebra 1 syllabus and agree with its content and commit to do what it is required of me in order to be a successful student.
Wednesday, August 10, 2011
Blogging is a very public activity. Anything that gets posted on the internet stays there. Forever. Deleting a post simply removes it from the blog it was posted to. Copies of the post may exist scattered all over the internet. I have come across posts from my students on blogs as far away as Sweden! That is why we are being so careful to respect your privacy and using first names only. We do not use pictures of ourselves. If you really want a graphic image associated with your posting use an avatar -- a picture of something that represents you but IS NOT of you.
Here are a few videos that illustrate some of what I want you to think about:
Two teachers in the U.S.A. worked with their classes to come up with a list of guidelines for student bloggers.
Students using blogs are expected to treat blogspaces as classroom spaces. Speech that is inappropriate for class is not appropriate for our blog. While we encourage you to engage in debate and conversation with other bloggers, we also expect that you will conduct yourself in a manner reflective of a representative of this school.
Never EVER EVER give out or record personal information on our blog. Our blog exists as a public space on the Internet. Don't share anything that you don't want the world to know. For your safety, be careful what you say, too. Don't give out your phone number or home address. This is particularly important to remember if you have a personal online journal or blog elsewhere.
Again, your blog is a public space. And if you put it on the Internet, odds are really good that it will stay on the Internet. Always. That means ten years from now when you are looking for a job, it might be possible for an employer to discover some really hateful and immature things you said when you were younger and more prone to foolish things. Be sure that anything you write you are proud of. It can come back to haunt you if you don't.
Never link to something you haven't read. While it isn't your job to police the Internet, when you link to something, you should make sure it is something that you really want to be associated with. If a link contains material that might be creepy or make some people uncomfortable, you should probably try a different source.
Look over the guidelines and add your own, if you like, in the comments section below this post. I think Bud's suggestions are excellent. We'll be using the one's I highlighted above as a basis for how we will use our blog. | 677.169 | 1 |
Wednesday, 30 April 2014
This post will be updated with links to solutions of all worksheets I
gave this week. It will be updated constantly, so keep a look out for
it.
REMINDER
- complete all your worksheets BEFORE you refer to the answers.
- grade your own worksheets and do the corrections.
- do not attempt to cheat as it will not help you in learning at all. SOLUTION SETS
Search for the following solutions (in italic) in GoogleSite. You can find all the answers under the respective headers highlighted in RED. Names of the files are italicized. HOMEWORK SOLUTIONS
Friday, 25 April 2014
Your assignment over the weekend is to complete the Common Test 2012 Paper. As this paper is slightly modified, do ensure that you keep to the following instructions. INSTRUCTIONS
- You are to complete the paper within 1 hour duration.
- Calculator is allowed.
- All necessary workings are to be shown clearly and in systematic manner.
- Try your best to not refer to any notes when attempting.
- Read the cover page carefully.
I will be doing intensive revision for all of you next week. Do take time off this weekend to revise through your examinable topics.
Wednesday, 23 April 2014
REVISION: FACTORIZATION
For your learning purpose, I have posted up the remedial worksheet on Algebra: Factorization. You are strongly encouraged to complete it in preparation of your upcoming Common Test.
You may download the worksheet via the following URL:
Tuesday, 22 April 2014
ASSIGNMENT
You are to complete the assignment on Introduction to Data Handling by Wednesday.
BAR GRAPH VS HISTOGRAM
Take some time to ponder on the difference between bar graph and histogram. The main question to ask is, "Under what circumstance(s) do you use bar graph or histogram to represent data?".
At the same time, these diagrams answer your questions to Unit 6.1 of your Data Handling notes. | 677.169 | 1 |
Pages
Schoolhouse Affiliate Link
Thursday, June 22, 2017
UnLock Algebra1 REVIEW
What perfect timing. As we leave the 7th grade and enter into the 8th grade we were given an opportunity to try out UnLock Algebra1 from UnLock Math. I was especially excited about trying this because it is an online video curriculum much like what we have been doing for math for the past few years. This curriculum is a complete program but can also be used as a supplement. It is accessible through a one-year subscription.
We have been using this program over the past month or so and find it pretty similar to our past math program. You log in online and there are parent and student accounts. From the Parent Login you are able to manage your student's account (assignment of courses, billing, payment info and passwords). You are also able to login to the Student Account from the Parent Login. If you give your student a password they can then login to their account by themselves. The program tracks progress and grades for you.
The lessons are broken down into units and there is a 472-page Complete Reference Notes document (for UnLock Algebra1) that you can refer to or printout that covers the material shown in the lesson videos. Each Unit consists of Daily Lessons and Review and Quizzes and Tests if applicable. A lesson consists of:
Warm up - to get your student in math mode
Short Video (less than 10 minutes)
Practice Problems - may be completed as many times as necessary and best grade will be recorded.
Stay Sharp problems for review
Challenge questions - can add 5% bonus on overall grade
Reference Notes - a link for that particular lesson
To start each unit the student clicks on the Unit to activate it then the Launch icon to begin that unit. This unlocks the unit and allows the student to begin lessons in that unit. Once lessons are complete they are graded and entered into the students record. There is also a Pacing Guide available as a pdf file to print or view online. It lists how many days it takes for each unit and the total days in the program. Each lesson is expected to take approximately 30 minutes (takes us closer to an hour). There are 170 days worth of lessons, reviews, and tests. If you work 4 days per week it can be completed in 43 weeks (10 3/4 months). If you work 5 days a week it would take 34 weeks (8 1/2 months).
The weighting of the grades is as follows:
Warm-ups - 0%
Practice Problems - 30%
Stay Sharp - 10%
Challenge - 5% bonus to overall grade but not penalized for bad grade
Review - 0%
Quizzes - 15%
Test - (can be taken multiple times with best grade recorded) 30%
Midterm - (can be taken multiple times with best grade recorded) 5%
Final Exam - (can be taken multiple times with best grade recorded) 10 %
The program tracks your student's progress in all areas. You can see if they are completing the warm ups, practice problems, stay sharp, and quizzes. I really like this program and think it is a good fit for how my son learns. The short video lessons are great for him. The lessons are not too long or cumbersome. The only challenge that I don't like compared to our old program is that he does not get immediate feedback on each problem. In UnLock Math when he answers a question he does not know until the end of the lesson what he got wrong or right. In the past this was a necessity for him (knowing immediately if he got problem right or wrong) just because of the way his mind works. But I can see a little maturity in him and it didn't seem to bother him as much as I thought it would. Score! Based on our experience we recommend this program for your consideration when looking for a math curriculum. You can connect with UnLock Math at FACEBOOK, TWITTER, or PINTEREST.
They also provide courses in Pre-Algebra, Algebra2 and now their newest release, Geometry! Be sure to check out the other reviews on UnLock Algebra1 and the other courses they offer by clicking on the banner below | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
4 MB|1,235 pages
Product Description
Higher Altitudes in Algebra I - Student Edition: Higher Altitudes in Algebra I presents and builds upon the fundamentals of beginning Algebra to prepare the student for the more sophisticated algebraic mathematics encountered in Algebra II. Comprehensive Algebraic concept coverage in expressions, arithmetic with polynomials and rational functions, creating equations, reasoning with equations and inequalities and problem solving through mathematical practices. | 677.169 | 1 |
Elementary Number Theory with Applications, Second Edition
This second edition updates the well-regarded 2001 publication with new short sections on topics like Catalan numbers and their relationship to Pascal's triangle and Mersenne numbers, Pollard rho factorization method, Hoggatt-Hensell identity. Koshy has added a new chapter on continued fractions. The unique features of the first edition like news of recent discoveries, biographical sketches of mathematicians, and applications--like the use of congruence in scheduling of a round-robin tournament--are being refreshed with current information. More challenging exercises are included both in the textbook and in the instructor's manual.
Elementary Number Theory with Applications 2e is ideally suited for undergraduate students and is especially appropriate for prospective and in-service math teachers at the high school and middle school levels.
* Loaded with pedagogical features including fully worked examples, graded exercises, chapter summaries, and computer exercises * Covers crucial applications of theory like computer security, ISBNs, ZIP codes, and UPC bar codes * Biographical sketches lay out the history of mathematics, emphasizing its roots in India and the Middle East
"synopsis" may belong to another edition of this title.
Review:
"This is the only number theory book to show how modular systems can be employed to create beautiful designs, thus linking number theory with both geometry and art. It is also the only number theory book to deal with bar codes, Zip coes, International Standard Book Numbers (ISBN), and European Article Numbers (EAN)...Each section provides a wealth of carefully prepared, well-graded examples and exercises to enahnce the readers' understanding and problem-solving skills." -ZENTRALBLATT MATH396212442754 940248 | 677.169 | 1 |
Showing 1 to 3 of 3
THE PRACTICES OF MONASTIC PRAYER:
ORIGINS, EVOLUTION, AND TENSIONS
Columba Stewart OSB
Saint Johns Abbey and University
Egyptian monks were known for the commitment to unceasing prayer, but what
did this really mean? This paper will explore forms of monas
LIMITS & CONTINUITY
LIMIT OF A FUNCTION
The concept of the limit of a function is the starting point of calculus. Without limits
calculus does not exist. Every notion in calculus can be expressed in some forms of
limits.
What is limit of a function? To un
Chapter 2.1 Practice Problems
EXPECTED SKILLS:
Be able to compute the average rate of change of a function over an interval; i.e., be
able to find the slope of the secant line through two points on the graph of a function.
Be comfortable using a limit t
Calculus I Advice
Showing 1 to 3 of 4
I would recommend the course. As someone who usually struggles through math, it seemed pretty straightforward. Notes are important, reviews are given for tests. The book wasn't necessary, but it's a pretty good resource if you need more examples of a problem, or for studying. The homework we had from the book was posted online. The professor was a little shy and quirky at times, but very nice and knowledgeable.
Course highlights:
This course began with limits. It's a concept that runs through half the course and carries into derivatives, and integrals. These are used to solve related rate problems, optimization problems, find the area under a graph, sketching a curve, and finding a tangent equation from a curve. I'm not sure how this relates to other aspects of calculus, or what this does as a whole. Since this is the only required math course I have, I doubt I'll be motivated to find out. But it was interesting nonetheless, and satisfying to see how easy everything is now looking back.
Hours per week:
3-5 hours
Advice for students:
For this course, it could have been enough to just take notes. If you want a good grade, really go for it and brush up on algebra, do practice questions, and write out a quick cheat-sheet of formulas and theorems and procedures. There are plenty of resources!
I have neutral feelings towards him because he does teach well and you will understand but he doesn't seem to answer your questions correctly. He doesn't let you use graphing calculators but you can use scientific calculators.
Course highlights:
You will learn about limits, derivatives, continuities and many other things.
Hours per week:
0-2 hours
Advice for students:
Pay attention to every single thing he talks about. Ask questions outside the box and for more difficult questions. | 677.169 | 1 |
A level maths coursework
AS and A Level: Maths. Browse by. Category: Core & Pure Mathematics (157). This coursework is about finding the roots of equations by numerical methods. (Original post by Alec) Within a maths module, what's the exam worth, and what's the coursework worth in terms of a percentage breakdown of the overal marks. Coursework in Mathematics: MEI discussion paper page 3 The current situation All GCSE Mathematics specifications contain two pieces of coursework, each worth. Numerical Solutions of Equations Mathematics Coursework (C3) Alvin Sipraga Magdalen College School, Brackley July 2009 1 Introduction In this coursework I will be. Imagine finding A level maths coursework help and A level physics coursework help service website that guarantees you absolute professionalism in handling your papers.
Study A-Level Mathematics from home. This course enables you to study for an A-Level in Mathematics by distance learning at. You study the same coursework. Coursework in Mathematics A discussion paper October 2006 Coursework in Mathematics: MEI discussion paper page 3 The current situation All GCSE Mathematics. Be part of the largest student community and join the conversation: Which A levels require absolutely NO coursework at all - 100% WRITTEN PAPER Exam. Coursework in Mathematics A discussion paper October 2006 Coursework in Mathematics: MEI discussion paper page 3 The current situation All GCSE Mathematics.
A level maths coursework
Study A-Level Mathematics from home. This course enables you to study for an A-Level in Mathematics by distance learning at. You study the same coursework. Frequently asked questions – Level 3, AS/A level Mathematics Which units have coursework?. The following units have coursework: Level 3 Certificate:. (Original post by Alec) Within a maths module, what's the exam worth, and what's the coursework worth in terms of a percentage breakdown of the overal marks. A-level Mathematics (6360) and A-level Statistics (6380) For use with the specifications from September 2004. The following tasks are recommended by AQA for centres. AS and A Level: Maths. Browse by. Category: Core & Pure Mathematics (157). This coursework is about finding the roots of equations by numerical methods.
Numerical Methods Coursework. Numerical Integration. The Problem. Integration means finding the area underneath a particular region of a function. At my current. Methods for Advanced Mathematics (C3) Coursework. Coursework for Methods for Advanced Mathematics. both in quantity and level of sophistication. Be part of the largest student community and join the conversation: Which A levels require absolutely NO coursework at all - 100% WRITTEN PAPER Exam.
A-level Mathematics (6360) and A-level Statistics (6380) For use with the specifications from September 2004. The following tasks are recommended by AQA for centres. Numerical Solutions of Equations Mathematics Coursework (C3) Alvin Sipraga Magdalen College School, Brackley July 2009 1 Introduction In this coursework I will be. C3 MEI A-level Maths Coursework. Uploaded by Maurice Yap. Spreadsheet Zero Of A Function Equations Mathematical Objects Elementary Mathematics. 0.0 (0) Download. Embed. Imagine finding A level maths coursework help and A level physics coursework help service website that guarantees you absolute professionalism in handling your papers. Coursework in Mathematics A discussion paper October 2006 Coursework in Mathematics: MEI discussion paper page 3 The current situation All GCSE Mathematics. | 677.169 | 1 |
Essential Education Math Instruction
Adult learners connect better with math when it is presented in relatable ways to solve everyday problems found in higher education, work, or life. This math program introduces basic numeracy concepts simple enough for ABE learners then advances through post-secondary college and career readiness levels.
Each math concept, from the most basic to algebra and geometry, is
related to adult learners' lives and familiar context. Adults
use math to manage finances, to complete home projects, and to make
decisions at work and home. Math is discussed in simple,
understandable, everyday language, with a focus on comprehension
and retention.
Call to Find Out More
Mathematics Content
Number Operations and Number Sense
Number operations and number sense are the foundation of math
skills. The GED Academy, TASC Prep Academy, and HiSET academy courses cover basic number knowledge;
addition, subtraction, multiplication, and division of whole
numbers, fractions, and decimals; problem solving; estimating;
rate, interest, money, percentages, ratios, and proportions;
and using calculators. These fundamentals give learners a solid
foundation for higher-level math and essential skills for daily
life. Mastery of number operations and number sense is critical
for learners with low-level math skills.
Measurement
This lesson introduces English and metric measurement systems and how to work with and convert units, while also addressing perimeter, volume and time. Measurement is an integral part of adult
learners lives at home, at work, and in school, and can be a
gap in adult learners' knowledge that is essential to
fill before moving forward.
Data Analysis, Statistics, and Probability
Data analysis, statistics, and probability requires being able
to think clearly about numeric information and analyze it in a
useful way. These skills will help students make decisions and
understand data in their financial and home lives as well as in
work and school. The data analysis, statistics, and
probability lessons cover graphs; mean, median, and mode;
charts and making conclusions from data; and probability and
statistics.
Geometry and Spacial Sense
Geometry and spacial sense is not only a critical math skill
but an important math arena for working with objects and
representations of space, from reading a map to redecorating a
room. Everyday geometry applications are addressed including lessons on perimeter and area, coordinate geometry, lines and angles, plane figures, volume and three-dimensional figures, and Pythagorean theorem.
Patterns, Functions, and Algebra
Pre-algebra to quadratic equations are all presented in an easy-to-understand format that teaches students skills to think through math problems logically. The
focus is on easy-to-understand language and the skills to think
through a math problem logically. Courses include
integers; exponents and scientific notation; variables,
expressions, and equations; problems with a missing element;
inequalities; linear equations; and quadratic equations.
Mathematics Test Familiarity and Practice
Students do better on the math portion of any high school equivalency test if they have some exposure to what they can expect before they take the test. Students learn time management skills and are introduced to mathematics test answer grids. Students also get practice with interactive question types such as text boxes, drop-downs and online calculators.
Follow UsLLC under license from the American Council on Education. Use of the GED trademark does not imply support or endorsement by ACE.
TASC Test Assessing Secondary Completion™ is a trademark of McGraw-Hill School Education Holdings LLC. They may not be used or reproduced without the express written permission from McGraw-Hill School Education Holdings LLC.
HiSET is the registered trademark of the Educational Testing Service. It may not be used without express written permission. | 677.169 | 1 |
MathUI 2016: The 11th Workshop on Mathematical User
Interfaces
Monday, July 25. 2016
Scope
MathUI is an international workshop to discuss how users can be
best supported when doing/learning/searching for/interacting with
mathematics using a computer.
Is mathematical user interface design a design for
representing mathematics, embedding mathematical context, or a
specific design for mathematicians?
How is mathematics for which purpose best represented?
What specifically math-oriented support is needed?
Does learning of math require a platform different than other
learning platforms?
Which mathematical services can be offered?
Which services can be meaningfully combined?
What best practices wrt. mathematics can be found and how can
they be best communicated? We invite all questions, that care for the
use of mathematics on computers and how the user experience can be
improved, to be discussed in the workshop.
Links
This workshop follows a successful series of workshops held at
the Conferences on Intelligent Computer Mathematics since more than 11
years; it features presentations of brand new ideas in papers selected
by a review process, wide space for discussions, as well as a software
demonstration session. | 677.169 | 1 |
emporary Geometry<br />Secondary students need more hands-on experiences with 3-D shapes.<br />Interdisciplinary by NatureEx. Biology and Art<br />Develop an appreciation in building and exploring models.<br />Technology is a must!<br />
4.
Prove It<br />Redefine the formal proofs of Euclidean geometry in the perspectives of synthetic, analytical, and transformational.<br />Synthetic Geometry: prove geometric relationships based on the use of a rational sequence of definitions, postulates and theorems.<br />Analytic Geometry: given a fundamental problem, one can find an equation to solve it (usually done through graphing technologies and computer software).<br />Transformational Geometry: explore geometric concepts through the use of transformational techniques (i.e. altering the parent graphs of y = x2 to be reflected and shifted).<br />
5.
Inductive and Deductive Reasoning<br />Inductive Reasoning: based on observations and collected data, generalizations are made.<br />Deductive Reasoning: based on accepted statements and a logical structure to reach a conclusion.<br />Contemporary Geometry can help introduce visuals for inductive and deductive reasoning and prepare students for formal proofs, with techniques such as mathematical induction.Ex. Linear Programming<br />
7.
21st Century Skills<br />The integration of a non-discrete algebra and geometry.<br />Technology can greatly assist students in a contemporary algebra where it is used today.<br />Technology can be: computers, software, probeware, graphing calculators, scientific calculators, or manipulatives.<br />
8.
Technology and Algebra<br />Solve systems of equations and inequalities, along with different types of equations such as linear, quadratic, etc.<br />Describe behaviors of multiple functions at once.<br />A need to model problems and demonstrate problem-solving skills through the use of technology for the current workforce, either after high school or college.<br />The goal is to present algebra to more students and can be approached in many different ways.<br /> | 677.169 | 1 |
Math
Please follow the link above to visit the Math Department Website for useful links and resources.
The Buchanan High School Mathematics Department consists of fourteen outstanding instructors committed to supporting ALL students in their mathematical development. The department offers courses that are in alignment with Common Core Standards and utilizes an integrated approach to mathematics by offering a series of courses that build upon each other (Math 1, Math 2, and Math 3). To support students, resources are available on the Math Department Website and are regularly updated. Our goal is to provide consistent first-time best instruction to ALL of our students! Throughout the year, after-school labs are available to support students with homework completion and test preparation. Teachers also provide office hours to assist students before and after school and during the school day. We encourage students to communicate regularly with their teachers and to seek out assistance early.
If you have questions about the sequences of courses offered at Buchanan High School please refer to the document below:
Math 3
Math 3 is the last of the three year integrated math sequences, here instructional time focuses on four critical areas: (1) apply methods from probability and statistics to draw inferences and conclusions from data; (2) expand understanding of functions to include polynomial, rational, and radical functions; (3) expand right triangle trigonometry to include general triangles; and (4) consolidate functions and geometry to create models and solve contextual problems. | 677.169 | 1 |
...
Show More mathematical situations.This prealgebra text, as a part of the Lial/Hornsby/Miller developmental worktext series, focuses on students by helping them reduce their confusion of symbols for subtraction with negative numbers. The text also clearly demonstrates why the product of two negative numbers is a positive number and explains why letters instead of numbers are used in algebra. This text helps students solve applied problems and increases their proficiency in solving fraction and percent | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
1 MB|11 printable pages
Product Description
A flip book is a simplified version of notes and examples for a larger concept. This (equations) flip book serves as your students' companion as they learn how to solve equations. It takes less than a period to cut out and make the books. Students use them for several weeks as they learn how to simplify expressions and solve equations. | 677.169 | 1 |
TEXT:
None required. See below.
Course Objective:
The goal of this course is to familiarize you with the basic tools
of mathematical physics. I will mix traditional
exposition of textbook material with modern applications.
These will mostly be drawn from condensed matter physics, since
I am most familiar with the mathematical physics
which is useful for current research in that field.
As an example, when we discuss matrix diagonalization and eigenvalues,
we will see how this can be used to compute energy bands in solids,
and unusual features like flat bands, disorder localized modes,
the 'Dirac' spectrum of graphene, etc
of interest in current research.
Homework assignments (due weekly on Fridays) will also contain
some numerical work. If the coding looks like it
will be too heavy, I will make those problems optional,
for extra credit. I plan to delay getting too
deep into numerical work
for a few weeks, so that I can take advantage
of material you will learn in Physics 102.
I have not selected a specific textbook. Most standard books
on mathematical physics
will have discussions of the material I present. I personally
like
"Mathematical Methods in the Physical Sciences",
by Mary Boas,
quite a bit. You will probably find a lot of useful initial material
for topics we will cover in your
second year calculus books, although we will likely go
a good deal further, and of course emphasize the physics
applications. | 677.169 | 1 |
Course Offerings
Middle School Math
Middle school mathematics emphasizes problem solving and assists the student in finding mathematical solutions. In addition to problem solving this course reviews and enhances the student's understanding of basic mathematical concepts. This class looks further into the order of operations, decimals, the metric system, basic geometry, data analysis, factors, proportions, ratio, integers, and algebraic equations.
Algebra
Algebra is the study of numerical patterns and their abstract representations. Students represent and analyze these patterns using functions, operations, tables, and graphs. Students learn these concepts and skills in a cognitively rich context of problem solving and critical reading and thinking. Advanced algebra concepts involve more complex reasoning and functions, conic sections and modeling.
Geometry
Geometry is the study of spatial reasoning. Students learn properties of two-dimensional and three-dimensional figures, classical reasoning, geometric construction, and Cartesian geometry. Students learn these concepts and skills in a cognitively rich context of problem solving and critical reading and thinking. Advanced studies in geometry can lead to trigonometry, the study of right triangles and their applications.
Natural Science
This course uses problem solving to introduce students to the scientific method and to assist their growth in an understanding of the nature of science. This course includes hands-on experiments and explorations to discover the physical and biological properties of earth systems, atmospheric systems, and natural systems. Through explorations of the natural areas around the school—the ponds, pastures, gardens, forests and sky—we will come to understand natural communities and to see human beings as integral and indispensable participants. Activities include field trips, scientific experiments and ecosystem studies.
Biology
This course uses problem-solving techniques to explore living systems. This course includes a variety of hands-on experiments and challenges students to explore basic life processes, cellular organization, mechanisms of inheritance, the dynamic relationship between organisms, and the change in organisms through time. As students explore these concepts they will be challenged to grow in their understanding of the scientific method and they will develop a greater sense of belonging to the community of life.
Chemistry
This course uses problem solving to introduce students to the interaction between matter and energy. This course includes many hands-on experiments and investigations involving laboratory equipment, basic elements, compounds, and mixtures. In this class students explore chemical reactions, writing chemical formulas, analyzing chemical equations, the gas laws, phase changes, and types of matter. The nature of this course requires significant understanding of basic mathematics principles and a good understanding of the scientific method.
Physics
Students will cooperatively investigate and understand the interrelationships among mass, distance, force, and time through mathematical and experimental processes utilizing experimental design techniques. Through analysis and interpretation of data, students will determine that quantities including mass, energy, momentum, and charge are conserved, but that energy can be transferred and transformed to provide usable work. Additionally, students will study wave phenomena including an understanding that different frequencies and wavelengths in the electromagnetic spectrum range from radio waves through visible light to gamma radiation. Students will be able to use the field concept to describe the effects of gravitational, electric, and magnetic forces, and to diagram, construct, and analyze basic electrical circuits and explain the function of various circuit components.
English
Middle school English classes strengthen students' reading ability and comprehension, composition skills, English usage, memory, critical and analytical thinking, and cultural awareness. As they study literature, students will focus on understanding and analyzing short prose passages and poetry using guided discussion and writing. Students will also complete independent reading of lengthier works, both classic and contemporary. Writing instruction emphasizes student achievement in style, organization, thinking, and usage through frequent informal and formal exercises in crafting sentences, paragraphs, essays, articles, letters, stories, and poems. Students will revise certain written assignments to further develop these skills. English grammar and usage are taught as tools for clear communication of meaning, both spoken and written.
In high school English, students will read more challenging passages, poems, and lengthier works. Literature instruction will continue to develop each student's ability to comprehend, recall, analyze, and enjoy prose, drama, and poetry. Composition assignments become more formal as stud organization, style, and usage. Students will often compose written reflections on assignments, discussions, and topics of interest. Improvement in content, style, and correct usage will be emphasized through required revision of certain long-term writing projects. Grammar instruction will focus on learning to identify and correct errors in English usage.
Latin I
This class emphasizes the study of Latin as a conduit for vocabulary and grammar skills building in English, Spanish, and other language studies. Study of Latin also provides a scaffold for further study of history, culture, science, and literature. Students will learn to analyze the structure of words and sentences as they attempt to translate and analyze meaning; from the first lesson, students will explore Roman culture and classical mythology while learning to read Latin.
Spanish Introduction and Spanish I
Introduction to Spanish focuses on teaching conversational Spanish and includes grammar, vocabulary and written Spanish. This class also introduces the student to Spanish and Latin-American culture. In Spanish I students begin to develop communicative competence in speaking, writing, comprehending and reading Spanish and expand their understanding of the culture of Spanish speaking countries. Our main goal in the class is to provide a context for everything we study. We give attention to grammar and vocabulary in a context of real-life situations, Latino cultures, and the language as a whole. The idea is to acquire a feel for the language, an appreciation of its beauty, and a degree of comfort and confidence in speaking it. Along the way, we learn a lot about the English language as well as Spanish.
Spanish II
The study of reading, writing, speaking, and listening continues as more complex grammatical structures are introduced. With more communicative competence come deeper discussions in Spanish about Latino culture and students' own experiences. Immersion in the language continues as students learn circumlocution in order to hold conversations entirely in Spanish.
Spanish III
The third year of Spanish continues development of listening, reading, writing, speaking, interpersonal communication and cultural awareness. Complexity and comprehension increase as students learn more vocabulary and grammar. Discussion in Spanish of Latino literature, culture, history and contemporary events is emphasized.
World Geography
World Geography class teaches students how to use maps, globes, atlases, satellite images, photographs, graphs, and other geographic tools to study and understand the world's populations, national identities and geographic environments. Students will look to history for understanding of how geological factors affect civilizations and cultures and economic, political, and social development. By the end of the class students will be able to identify all the continents, oceans, many islands and nations, and understand that the world's population is a single community divided by geographic location.
World History
The World History Part I course covers ancient civilizations. From the beginning of man's existence on the planet, through the evolution of villages into city-states and the cradle of civilization in the Fertile Crescent, we explore how man has interacted with his environment, set-up communities, built structures, and established governments. This course takes us around the globe to sites like the Roman walls throughout Europe, the Great Wall of China, and the sacred rivers of India. Students will gain comprehension of the history which led to governments, religions and wars, and which established the foundation of the Old World.
The World History Part II course is the second year of a survey of world history. Students will engage in a study of the political, cultural, social, and economic conditions from 1500 to modern times. Students will engage in study and analysis of important world events such as the Reformation, the Age of Discovery, the Industrial Revolution, World War I, World War II, the Cold War, and the independence movements of the 20th century. We will study the impact of these events on global trade, science, politics, economics and religion around the world.
U.S. History & Government
We will study the original American's habitat and civilizations before the arrival of Europeans. Students will explore the early history of the United States to understand the ideas and hard work that built the union. They will also study documents, speeches and the constitution that laid the foundation of American ideals and institutions. The study of history emphasizes the intellectual skills required for responsible citizenship. The student must demonstrate understanding of the essential knowledge of the history of the United States of America.
Art
Using traditional techniques and emerging technologies, this class emphasizes exploration, analysis, and investigation of the creative process. Students develop technical skills that empower them to communicate ideas visually, developing an understanding of and appreciation for the visual arts. Students explore various two-dimensional and three-dimensional art media, using different expressive and technical approaches. Students study the factors that distinguish artistic styles and that clarify the role of art in culture.
Instrumental Music
The instrumental music program emphasizes basic musical skills for band and orchestra. Students study the elements of music, and how to create and to enjoy music.
Physical Education
The physical education curriculum includes sports such as softball and kickball, which are played during the physical education class period, as well as hiking, swimming, canoeing, and skiing, which are done as group outings. Students can choose to participate in horseback riding lessons during the physical education class period.
Farm Chores
Our daily chores sustain our farm, our school and our community. We all take part in this work twice a day: in the morning and again in the afternoon. General farm chores, gardening, keeping our building clean, kitchen duty, and harvesting from the farm and garden are among the activities at these times. In the process of doing this work, we learn about the needs of animals, about health and nutrition, how to care for trees and vegetable plants, how to maintain a household, and many other skills that we will rely on throughout our lives whatever our living situation. Students should come prepared to work outside every day in all weather conditions. A hat, good work gloves and boots are essential items that are kept at school. | 677.169 | 1 |
HONORS MATH 3 COURSE INFORMATION
Honors Math III is a course that will include topics from geometry, algebra, statistics and probability. It is the third course in the Common Core series of mathematics—at the honors level, the expectation is that students are comfortable with the topics presented in Math I and II because there is little time allotted for review. Students in this course should be strong mathematics students who finished Honors Math 2 with an A or B and who look forward to a challenging, rigorous mathematics experience. | 677.169 | 1 |
Showing 1 to 23 of 23
Part 1
Foundations
Chapter 1
Sets and vector spaces
We assume some familiarity with basic notions from set theory and linear
algebra. For example the reader should be comfortable working with nite
dimensional vector spaces over a eld, their bases, and lin
Calulus
Rie Mathematis Tournament 2000
1. Find the slope of the tangent at the point of inetion of y = x3
3x2 + 6x + 2000.
2. Karen is attempting to limb a rope that is not seurely fastened. If she pulls herself
up x feet at one, then the rope slips x3 fe
Harvard-MIT Mathematics Tournament
February 19, 2005
Team Round A
Disconnected Domino Rally [175]
On an infinite checkerboard, the union of any two distinct unit squares is called a (disconnected) domino. A domino is said to be of type (a, b), with a b in
Harvard-MIT Mathematics Tournament
March 15, 2003
Individual Round: Calculus Subject Test
1. A point is chosen randomly with uniform distribution in the interior of a circle of radius
1. What is its expected distance from the center of the circle?
2. A pa
Harvard-MIT Mathematics Tournament
February 19, 2005
Team Round B
Disconnected Domino Rally [150]
On an infinite checkerboard, the union of any two distinct unit squares is called a (disconnected) domino. A domino is said to be of type (a, b), with a b in
HMMT 1998: Calculus Solutions
1. Problem: Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks
a sinusoidal path in search of home; that is, after t minutes he is at position (t, sin t).
Five minutes after he sets
Harvard-MIT Mathematics Tournament
March 15, 2003
Individual Round: Calculus Subject Test Solutions
1. A point is chosen randomly with uniform distribution in the interior of a circle of radius
1. What is its expected distance from the center of the circl
Power Question - Coloring Graphs
Perhaps you have heard of the Four Color Theorem (if not, dont panic!), which essentially says that any
map (e.g. a map of the United States) can be colored with four or fewer colors without giving neighboring
regions (e.g
Algebra Midterm 1
Fall 2012
Work out if each of the following statements is TRUE or FALSE.
Justify your answer carefully by supplying a PROOF or a COUNTEREXAMPLE.
1. Let Vecf pRq be the category of nite dimensional vector spaces
over R, and D : Vecf pRq V
Modules review
True or False?
1. Let V W X and V W Y be decompositions of a left
R-module V as direct sums of submodules. Then X Y .
2. Let V W X and V W Y be decompositions of a left
R-module V as direct sums of submodules. Then X Y .
3. If V is a simple
Sample homework
solutions 1
1.2.3 Let V be a nite dimensional vector space and W V be a subspace. Show that W is nite dimensional and any basis of W can be
extended to a basis of V . Deduce that dim W dim V with equality
if and only if W V .
If W is not n
Algebra: Assignment 5
Due on Firday, November 2, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Contents
Exercise 4.4.10
Part (a) . . .
Part (b) . . .
Part (c) . . | 677.169 | 1 |
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Are you looking for Classics Of Mathematics Books? You can Download and Read OnlineClassics Of Mathematics Book for Free.
You can see the list of related books. Click on Download or Read Online button to get the full book.
Appropriate for undergraduate and select graduate courses in the history of mathematics, and in the history of science. This edited volume of readings contains more than 130 selections from eminent mathematicians from A `h-mose' to Hilbert and Noether. The chapter introductions comprise a concise history of mathematics based on critical textual analysis and the latest scholarship. Each reading is preceded by a substantial biography of its author.
Appropriate for undergraduate and select graduate courses in the history of mathematics, and in the history of science. This edited volume of readings contains more than 130 selections from eminent mathematicians from A h-mose' to Hilbert and Noether. The chapter introductions comprise a concise history of mathematics based on critical textual analysis and the latest scholarship. Each reading is preceded by a substantial biography of its author.
What makes for a philosophical classic? Why do some philosophical works persist over time, while others do not? The philosophical canon and diversity are topics of major debate today. This stimulating volume contains ten new essays by accomplished philosophers writing passionately about works in the history of philosophy that they feel were unjustly neglected or ignored-and why they deserve greater attention. The essays cover lesser known works by famous thinkers as well as works that were once famous but now only faintly remembered. Works examined include Gorgias' Encomium of Helen, Jane Adams' Women and Public Housekeeping, W.E.B. DuBois' Whither Now and Why, Edith Stein's On the Problem of Empathy, Jonathan Bennett's Rationality, and more. While each chapter is an expression of engagement with an individual work, the volume as a whole, and Eric Schliesser's introduction specifically, address timely questions about the nature of philosophy, disciplinary contours, and the vagaries of canon formation.
As an historiographic monograph, this book offers a detailed survey of the professional evolution and significance of an entire discipline devoted to the history of science. It provides both an intellectual and a social history of the development of the subject from the first such effort written by the ancient Greek author Eudemus in the Fourth Century BC, to the founding of the international journal, Historia Mathematica, by Kenneth O. May in the early 1970s.
This compact, well-written history covers major mathematical ideas and techniques from the ancient Near East to 20th-century computer theory, surveying the works of Archimedes, Pascal, Gauss, Hilbert, and many others. "The author's ability as a first-class historian as well as an able mathematician has enabled him to produce a work which is unquestionably one of the best." — Nature.
The 8th edition of Classics of Western Philosophy trumps the 7th edition¿as well as all competing anthologies¿by including Plato¿s Laches, selections from Aristotle¿s Nicomachean Ethics (on courage), Descartes¿ Discourse on Method, the remainder of Berkeley¿s Treatise Concerning the Principles of Human Knowledge, and the whole of Kant¿s Prolegomena, all in preeminent translations. These additions¿with no offsetting deletions¿yield an anthology of unprecedented versatility, making it the only volume of its kind to offer both Descartes¿ Discourse on Method and Meditations on First Philosophy; Berkeley¿s A Treatise Concerning Principles of Human Knowledge and Three Dialogues; and Kant¿s Prolegomena and selections from the Critique of Pure Reason. | 677.169 | 1 |
Description
Passing grades in two years of algebra courses are required for
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Second Year Calculus: From Celestial Mechanics to Special Relativity covers multi-variable and vector calculus, emphasizing the historical physical problems which gave rise to the concepts of calculus. The book guides us from the birth of the mechanized view of the world in Isaac Newton's Mathematical Principles of Natural Philosophy in which mathematics becomes the ultimate tool for modelling physical reality, to the dawn of a radically new and ofte
This book offers an essential bridge between college-level introductions and advanced graduate-level books on special relativity. It begins at an elementary level, presenting and discussing the basic concepts normally covered in college-level works, including the Lorentz transformation.
by Riccardo D'Auria (Author), Mario Trigiante (Author) Second revised edition Enables the reader to understand the concepts that form the foundational of contemporary theoretical particle physics Places particular emphasis on the role of symmetry in modern theoretical physics | 677.169 | 1 |
Welcome to Math for Teachers!
This is the course homepage and syllabus for Math 2137,
Algebra and Coordinate Geometry for Teachers. Here you will
find our course calendar, current assignments, basic course
information, and links to additional content of interest. This
syllabus may need to be updated as the course progresses, and you will
always find the current version
at
This boy is using geometry to understand algebra and algebra to
understand geometry.
The topics of middle-grade math bring adventure and excitement!
Together, Math 2137 and 2138 will hone your skills of mathematical
explanation and explore the profound relationships between algebra and
geometry.
Course Objectives
This course integrates the various types of numbers introduced in the
previous course to present them as members of a single (real) number
system. The notion that new numbers are discovered as solutions to
equations is promoted, and motivated by connecting various equations
with mathematical models.
Matrices are introduced and used as linear transformations, mainly in
the plane. The complex numbers are introduced as general
solutions to quadratic equations and the relationship between complex
arithmetic and transformations in the plane is explored.
The course finishes with several weeks of geometry content for
middle grade teachers, including more material on proofs, triangle
congruence, and non-Euclidean geometry. The main example is "Taxicab
geometry", based on the ell_1 norm.
Topics
Polynomial arithmetic as "base-x" and binomial theorem
Real number system
Polynomial equations and their roots
Exponential and logarithm functions
Complex numbers
Matrices
Complex arithmetic and linear transformations in the plane
Geometry proofs
Taxicab geometry
Learning Goals
Understand polynomial arithmetic from the perspective of place value.
Unified perspective on the real number system, including
situations modeled by different numbers, and numbers as solutions to
equations.
Familiarity with complex numbers and matrices from algebraic and
geometric points of view.
Awareness of non-Eulidean geometries and the importance of the
parallel postulate.
Ability to create and evaluate geometric proofs.
Identify major historical developments in algebra and number systems
including contributions of significant figures and diverse cultures.
Homework 8
Due Monday, 11/07:
Consider the 2 x 2 matrices A and B below. Let I be the
identity matrix, also below.
\[
A = \begin{bmatrix}3 & 1\\2 & 4\end{bmatrix} \quad
B = \begin{bmatrix}p & q\\r & s\end{bmatrix} \quad
I = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}
\]
Explain how the equation AB = I can be viewed as a system of four
linear equations with four variables. Solve this system of equations.
Solve exercises 2.1.4, 2.1.9(b) both directly, and with the
determinant formula. Explain how the two methods are related.
Consider a square with corners labeled (in counter-clockwise
order) 1, 2, 3, 4. Let r be the permutation (1 2 3 4) and
let s be the permutaiton (2 4). Note that composition of
permutations is not generally commutative, but it is associative.
Write all the different permutations you can get by combining
r and s. (There should be 8 distinct
possibilities; combining any two yields one of these 8, and you can
show all of these on an 8 x 8 grid like a multiplication table.)
Compare with exercises 9 -- 12 in section 6.4.
Homework 7
Due Monday, 10/31:
17.1.4, 7.1.7, 17.2.1, 17.3.5, 17.4.3.
Homework 6
Prepare for student presentations. A list of main topics is due
Monday, 10/17. Presentations will be the following week: 10/24 and
10/26.
Subjects are:
Complex numbers (Ch. 15)
Hyperbolas (Sec. 12.4, 12.5)
Permutations (Ch. 14; mostly sec. 14.3)
Homework 5
Due Monday, 10/17:
13.1.8, 13.2.1, 13.2.3, 13.2.8(a), 13.2.8(d), [two problems of your
choosing from 13.3], 13.4.6, 13.5.5.
Additional 1: Let a be a real number greater than 1, and
let f be a function. Discuss the differences between the
graphs of f(x), a * f(x),
and f(a*x). Give one or two illuminating examples.
Additional 2: Let f(x) = 2x^3 - 3x^2 - x + 4. Use the
method outlined in class to rewrite f as a function
of (x-c) for c = 3 and c = -1.
Homework 4
Due Monday, 09/26:
1. Find the center and radius for the circle described by
\[
x^2 + 4x + y^2 - 2y - 4 = 0
\]
Hint: complete the squares
2. Find 3 Pythagorean triples that we have not ever discussed in
class, and are not multiples of ones discussed in class.
3. Find 3 rational points on the circle from part 1. Hint: use
translation and dilation from the unit circle.
4. Consider a parallelogram with side lengths 3 and 5 and with short
diagonal 4. What is the length of the long diagonal?
10.2.1, 10.2.5
Homework 3
Due Monday, 09/12:
Prepare class presentations! Plan for 20 -- 30 minutes.
In addition to your presentation,
choose a few helpful homework problems to assign to the other students.
Homework 2
Due Wednesday, 09/07:
3.2.1, 3.2.3, 3.4.4, 3.4.8, 3.4.13, 3.4.19, 3.4.27, 3.4.29
Additional Problem: Show that \(\mathbb{Q}(\sqrt{5})\) satisfies the
additivity and multiplicativity axioms for
number systems (the most interesting one is multiplicative inverses).
Also: Prepare a 20 minute presentation on the sections you volunteered
for in class! Either 8.3/8.4, Ch. 4, or 9.1/9.2.
Basic information
Instructor
Textbook
The style of this book is different from typical math textbooks,
and is much closer to the way mathematicians communicate with
eachother. Lang invites the student to experience and learn
mathematics at a level which is deep, but not complex. Our focus
will be Parts III and IV, but it is illuminating to compare the
first two parts with Beckmann's text.
Additional Texts:
Bart Snapp has developed free texts for middle
grade teachers, and they can be a helpful supplement to Lang.
Midterms (2In-class participation: 20%
Homework: 20%
Midterm exams: 20% each, for a total of 40%
Final exam: 20%
Participation
Thoughout this semester we will be focused on the how
and why of basic know | 677.169 | 1 |
Online High School Courses
There are many reasons to consider online high school courses. Whether you are looking for online summer school courses, looking to re-enroll after dropping out of traditional high school, or want to refresh your memory on challenging topics, Penn Foster High School can help. We offer a variety of online high school courses for you to choose from. Our courses are affordable, flexible, and allow you to learn on your own time, at your own pace, at home or on the go.
A review of basic math skills and principles along with a study of various business math topics such as income, maintaining a checking account, interest, installment buying, discounts, and markups.
Calculus (1 credit)
Explains the derivative of a function and the applications of derivatives, the integral and how to use it, and methods of integration.
Consumer Math (1 credit)
Learn how math applies to everyday life - including lessons on money, employment, purchases, insurance, savings and investments.
Consumer Math Advanced (1 credit)
This two-part course teaches students tools and concepts related to everyday life, including income tax, loans, credit cards, money management, shopping for a car or a home. It also includes sections on insurances and investments, as well as business data analysis, managing people and inventory and sales and marketing.
General Math I (1 credit)
Learn fundamental operations including whole numbers, decimals and fractions, and advances into ratios, proportions and percentages.
General Math II (1 credit)
Learn fundamental operations including whole numbers, decimals and fractions, and advances into ratios, proportions and percentages.
Geometry (1 credit)
Study points, lines, planes and angles, and more advanced topics like polygons, triangles, circles and solids.
Geometry Advanced (1 credit)
This two-part course provides hands-on activities to help students understand the theoretical relationship between lines, angles, polygons, circles and three-dimensional figures.
Pre-Algebra Advanced (1 credit)
This two-part course will cover rational numbers and proportional relationships, as well as ratios, similarity and using percents to solve problems.
Business Software Applications Advanced (1 credit)
A study of basic operations with signed numbers, monomials, and polynomials. Also includes formulas, equations, inequalities, graphing, exponents, roots, quadratic equations, and algebraic fractions.
Computer Applications Advanced (.5 credit)
This course will teach basic techniques used for designing webpages and will show you how to insert text, tables, pictures and multimedia files. In this course, you'll create three website projects that you designed yourself.
Microsoft Office Advanced (.5 credit)
This course provides step-by-step tutorials to teach the basic features of Microsoft Word®, PowerPoint®, and Excel®.
Microsoft Word & Excel (1 credit)
Learn how to use Microsoft Word® to create, edit and illustrate documents. In addition, you'll cover basic features in Microsoft Excel®.
Small Business Management (1 credit)
Learn how to start your own business, including the basics of developing a business plan.
Understanding Computers Advanced (.5 credit)
Learn computer basics, using technology for research and problem solving, and the security risks in using computers.
American Literature (1 credit)
Study literary terms, structural elements of literary genres and learn to interpret literature for knowledge and enjoyment. Learn what it has meant to be an American during each major period of American Literature.
American Literature Advanced (1 credit)
In this two-part course you'll cover American literature ranging from Native American poems and colonialists' observations of the New World to American literature through World War II to today.
Basic English (1 credit)
Learn the fundamentals of the English language to help you effectively communication. This course covers capitalization, punctuation, grammar, and spelling.
English Communication (1 credit)
Learn how to avoid grammatical errors when writing sentences and paragraphs, and how use words to effectively communicate.
Grammar Advanced (1 credit)
In this two-part course, you'll learn the fundamentals of grammar and then move onto to learning the dominant structure of a paper, as well as how to write using engaging, persuasive and informative writing styles.
Grammar & Literature Advanced (1 credit)
In this two part course, you'll learn how to master essential reading and writing skills. You'll learn basic rules of writing, as well as grammar, punctuation, capitalization and spelling.
Literature (1 credit)
Read short stories, nonfiction, poetry and drama to get a deeper appreciation of the relationship between literature and life.
In this two-part course, you'll learn basic literary elements like plot, setting, character and conflict, and also explores the use of imagery and symbolism to convey meaning in writing. You'll learn how to analyze and compare works of the same genre, as well as read and interpret popular media and workplace documents.
Practical English (1 credit)
In this course you'll develop your writing skills to learn the importance of sentence structure, paraphs, letters and compositions. The course begins with the study of grammar and then focuses on the writing process and emphasizes an individual approach to writing.
World Literature Advanced (1 credit)
In the first part of this two part course, you'll learn about writings from thousands of years ago from ancient Middle East, Greece, Rome, Africa and Asia. In the second half of this course you'll cover writings from the Middle Ages through modern times.
Written Communication (1 credit)
This course will provide you with step-by-step instructions for handling sentence structure, punctuation, pronunciation, paragraphing and grammar.
American History (1 credit)
Learn about people, events and sociopolitical forces that have shaped American, from the discovery of the continent to present times. See how American history plays a role in today's events and global conditions.
American History Advanced (1 credit)
In the first part of this two-part course, you'll cover the the discovery and settlement of the country through the late nineteenth century. The second half of this course covers time periods from the end of the nineteenth century to the present.
Civics (1 credit)
This course covers the rights, freedoms and responsibilities of American citizens. Learn about the roots of American government and how it operates today. You'll also cover the relationships between American and other nations.
Civics Advanced (1 credit)
In the first part of this two-part course, you'll learn about the foundation of the American constitutional democracy, how the US government is organized and the role of citizens in government. The second half of this course will explore the role of citizens in society, the US economic system and foreign policy.
Economics (1 credit)
In this course, you'll learn about the different economic systems across the world. You'll learn the function of money, the law of supply and demand and the role of banks and governments in capitalist economies.
Economics Advanced (1 credit)
In the first part of this two part-course, you'll focus on microeconomics and how businesses move resources around the economy to produce products for sale to households, foreign marketings, governments and other businesses. The second half of this course will focus on macroeconomics, which explores how consumers, businesses, and foreigners impact the total amount of output, employment and income for the economy.
Geography Advanced (1 credit)
In the first part of this two-part course, you'll learn about maps, weather, bodies of water, population, cultural geography, and explores North and South America and Europe. In the second half of this course will cover Russia, Northern Eurasia, Africa, Asia, and Pacific islands.
Psychology (1 credit)
This course provides an introduction into the roots and development of modern psychology. You'll learn about states of consciousness, and theories of intelligence, development, and personality.
Psychology Advanced (1 credit)
In the first part of this two-part course, you'll explore how psychologists unravel the mystery of what it is to be human, the biological processes that enable us to make sense of the world around us. The second half of this course, covers emotions, their components and how they affect our lives. You'll also review motivation, human growth and development and gender roles.
World Geography (1 credit)
You'll learn about the physical environment of human beings and the means they use to supply their political and economic needs. Includes maps, charts, graphs, and pictures of the main areas of the world.
World History Advanced (1 credit)
In the first part of this two-part course, you'll learn about ancient civilizations in Africa, Europe and Asia through the Middle Ages and the Renaissance. The second half of this course will cover the Enlightenment, the Age of Imperialism, Industrialization, the wards of the 1900s, through present times.
Biology (1 credit)
In this course, you'll learn about the characteristics of life, chemistry of cells, and the links between life and energy. You'll also learn about the theory of evolution, ecosystems, heredity, and adaptation.
Biology Advanced (1 credit)
In this two-part course, you'll learn about the characteristics of life, including evolution and homeostasis.You'll also learn the defining characteristics of organisms belonging to different taxonomic groups, and cover topics like bacteria, protists, fungi, viruses, and the form and function of plants and animals.
Chemistry (1 credit)
You'll learn about the study of structure and reactions of matter. This course also covers the elemental symbols, chemical reactions and the role of energy in those reactions.
Chemistry Advanced (1 credit)
In the first part of this two-part course, you'll learn about the science of chemistry, matter and energy, atoms and moles, the periodic table, and more. In the second half of this course you'll learn about states of matter and intermolecular forces, gases, solutions, chemical equilibrium, acids and bases, reaction rates electrochemistry and more.
Earth Science (1 credit)
In this course, you'll learn about the scientific method, the formation of the solar system, the moon's phases, movement of earth, plate tectonics, the formation of the oceans, and erosion. You'll also learn about rock and mineral analysis, soil formation, weather patterns and chemical principles.
Earth Science Advanced (1 credit)
In this two-part course, you'll learn about the structure, composition and natural processes of Earth. You'll also learn to analyze rocks, student earthquakes and volcanoes, examine fossils, explore the oceans, and even learn how to predict the weather.
General Science (1 credit)
You'll learn the basics of a variety of the sciences, including physics, chemistry and biology. This course covers atoms and molecules, light and sound, electricity and magnetism, astronomy, the rise of life on Earth, human anatomy and genetics.
Physical Science (1 credit)
In this course, you'll learn about matter and energy - their nature and the relationship between them. You'll cover topics like water, the chemistry of building materials, fuels, natural and synthetic rubbers and plastics, and energy in relation to motion and force.
Physical Science Advanced (1 credit)
In this two-part course, you'll cover both physics and chemistry. You'll study the states and structures of matter and also study heat and temperatures, waves, sound and light. You'll also be introduced to earth and space science.
Physics Advanced (1 credit)
In this two-part course, you'll learn about the science of physics. Ranging from motion in one and two dimensions, vectors, forces and the laws of motion. You'll also learn about light, reflection, refraction, electricity, and magnetism.
Art Appreciation (1 credit)
In this course, you'll be introduced to various forms of art throughout history, from prehistoric to modern. You'll also learn to evaluate the meaning and quality of individual works.
Art Appreciation Advanced (.5 credit)
This course will provide you with a basic knowledge of the history, media, techniques, tools and cultural implications of the visual arts.
Human Relations (1 credit)
Learn different methods of analyzing and improving relations with other people in personal life and in working environments. This course focuses on individual productivity, teamwork, working relationships, and dealing with frustration.
Music (1 credit)
In this course, you'll learn about different forms of music throughout history, from medieval times to present day. You'll also learn about music theory and instrumentation.
Music Appreciation Advanced (.5 credit)
In this course, you'll learn basic music theory, analysis and history in order to broaden your interest in and understanding of music.
Reading Skills (1 credit)
This course will teach you the techniques applicable to any type of reading. This includes reviewing, predicting, scanning, finding the main idea and drawing conclusions.
Spanish (1 credit)
In this course, you'll learn how to speak and comprehend the Spanish language. You'll learn Spanish vocabulary, grammar and this course will help you improve your fluency through listening to and creating stories.
Spanish I Advanced (1 credit)
In this two-part course, you'll learn the fundamentals of Spanish vocabulary and grammar through reading, writing and oral activities. You'll also learn vocabulary important for ordering in a restaurant, cooking meals, celebrating holidays and traveling to Spanish speaking countries.
Spanish II Advanced (1 credit)
This course is a continuation of Spanish I Advanced. You'll continue to learn new words, phrases and verb tenses. You'll also study customs and traditions in different Spanish-speaking cultures. | 677.169 | 1 |
Math 1910, Section 9, Test #4 (Chapter 4)
Name_
Each of these 10 problems has a value of 5 points.
1. Find all the critical numbers of each of the following functions:
(a) f(x) = x4(x-3)3
Use the product rule to calculate f'(x). The critical numbers are 0
Math 1910, Section 9, Test #2 (Chapter 2)
Name_
Each of these 15 problems has a value of 5 points.
1. Use the precise definition of a limit to prove that lim ( 97x )=5 . The first statement of your
x 2
proof has been written for you to help get you starte
Math 1910 Test #1 solution guide
1. Computing the domain means finding the values of x for which the value of the expression can
be computed. The things to watch out for are zeros in denominators and negative valuesunder a
square root. The trouble points
Math 1910, Section 9, Fall 2011: Extra Credit Quiz #7
Math 1910, Section 9, Fall 2011: Extra Credit Quiz #7
Name_
Name_
Use l'Hpital's rule to help you compute each of the following
0
limits. The first expression has the form
and the second has
0
the form
Math 1910, Section 9, Fall 2011: Extra Credit Quiz #5
Name_
The level of interest in Professor Wiggen's lectures is a function of time, I(t), which wanes at a rate
which is proportional to the current level of interest. Students begin each class period wi
Calculus I Advice
Showing 1 to 3 of 9
This professor has a very straightforward teaching style - lecture and quiz, with some tests thrown in. Everything was taught clearly and easy to understand.
Course highlights:
I gained experience in how college works. The class material was easy since I had taken Calc 1 in high school, but I wasn't used to the "college style" way of teaching. This has helped with much of my college career so far.
Hours per week:
3-5 hours
Advice for students:
If you don't understand, DON'T LET IT DROP!! If you fall behind even a little, it will snowball on you. You may have heard this before, but if you don't pay attention, it can happen - even to a straight A high school student. (Personal experience here.)
Course Term:Fall 2016
Professor:JaredC.Daniels
Course Required?Yes
Course Tags:Math-heavyMany Small AssignmentsA Few Big Assignments
May 11, 2017
| No strong feelings either way.
Not too easy. Not too difficult.
Course Overview:
This would be a good course to take for anyone to take going into business
Course highlights:
I learned that putting time into this class is vital to making a good grade
Hours per week:
0-2 hours
Advice for students:
Studying is required for this class. It will really help with your grades
I would recommend this course because it helped plant a strong foundation for my future Mathematics classes in college. All of Calculus hinges on the ideas taught in this class.
Course highlights:
I learned a great basic knowledge of Calculus. We went over limits, derivatives, and integrals. Limits will be used in every area of calculus. It is what allows us to solve problems that can't be solved the traditional way.
Hours per week:
3-5 hours
Advice for students:
Coming into this course, you need to know that you will need to study and complete all of the homework. Most professors only put problems from the homework on the quizzes and tests. Our class was mostly the professor assigning homework and the class would be the questions we didn't understand. It was not very lecture-heavy. It was very interactive. Also, you need to have pre-calculus knowledge before taking this class. Trigonometry will be used often. | 677.169 | 1 |
Welcome to the Algebra2go® allied health page. These resources were created to assist Saddleback College's allied health students prepare for chemistry 108. At this time we can offer two types of study materials: video lectures and video worksheets. Whichever materials you decide to use, we would appreciate your feedback. Please let Professor Perez know what you liked and didn't like about these materials by sending him an e-mail.
Video Worksheets
Every video is accompanied by a worksheet that mirrors the presentation so that you can work along with Charlie during the video. This kind of active learning process is a particularly good way to increase retention. The worksheets are saved in the .pdf format. If your computer doesn't already read these files, you can download the free Adobe Acrobat Reader.
Video Lectures
Perhaps our most popular tools are the video lectures. Attend Professor Perez's virtual classroom alongside his favorite student, Charlie. Each lecture focuses on a particular skill set which is vital for success. We are in the process of adding closed captioning to all of the videos. For a few videos we also have versions with an ASL interpreter. We hope to have more of these in the future. The videos are available in three formats.
Windows Streaming Media:
These are .wmv videos. If your computer doesn't already play this format, you can download the free Windows Media Player to watch them. Mac users might find it easier to install the free Flip4Mac components for Quicktime.
YouTube:
These are flash videos. If your computer doesn't already play this format, you can download the free Adobe Flash Player to watch them. YouTube videos can be played on a wide range of platforms such as computers, iPhones, Wii's, etc., but because of limitations on the length of these video clips some of the videos have had to be split into multiple parts. YouTube has recently added a feature which will translate the closed captioning into several dozen languages.
TeacherTube:
These are flash videos similar to the YouTube versions, but TeacherTube videos do not need to be split into multiple parts. If your computer doesn't already play this format, you can download the free Adobe Flash Player to watch them. | 677.169 | 1 |
Hi All, I am in need of assistance on equivalent fractions, trigonometric functions, y-intercept and reducing fractions. Since I am a beginner to College Algebra, I really want to learn the basics of Remedial Algebra fully . Can anyone recommend the best resource with which I can start learning the fundamental principles ? I have a midterm next week.
Aaah May Jesus save us students from the wrath of how to use a scientific calculator calculations. I used to face same problems that you do when I was there. I always used to be confused in Basic Math, Pre Algebra and Pre Algebra. I was worst in how to use a scientific calculator calculations till I came to know of Algebrator. It is really Great and I would surely recommend it. The best feature of the software is that it will also help you learn algebra and not just provide your answers. I found Algebrator effective and am sure it will help you too. Cheers.
Algebrator is a beneficial tool .Algebrator is the program that I have used through several math classes - Algebra 1, Basic Math and Pre Algebra. It is a really a great piece of algebra software. I remember of going through problems with leading coefficient, quadratic formula and trigonometry. I would simply type in a problem homework, click on Solve – and step by step solution to my algebra homework. I highly recommend the program. | 677.169 | 1 |
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Once again, bestselling author and award-winning teacher Andy Field hasn't just broken the traditional textbook mould with his new novel/textbook, he has forged the only statistics book on the market with a terrifying probability bridge, zombies and a talking cat! Andy Field's unique approach gently introduces students across the social sciences to the importance and relevance of statistics in a stunningly illustrated format and style. By weaving in a compelling narrative, he takes students on an exciting journey through introductory level statistics overcoming potential anxiety around the subject and providing a vibrant alternative to the dullness of many typical offerings. The medium, the message and the rock-solid statistics coverage combine to raise the level of attainment of even the most Maths-phobic student. It assumes no previous knowledge, nor requires the use of data analysis software. It covers the material you would expect for an introductory level statistics module that his previous books (Discovering Statistics Using IBM SPSS Statistics and Discovering Statistics Using R) only touch on, but with a contemporary twist, laying down strong foundations for understanding classical and Bayesian approaches to data analysis. In doing so, it provides an unrivalled launchpad to further study, research and inquisitiveness about the real world, equipping students with the skills to succeed in their chosen degree and which they can go on to apply in the workplace. Our Facebook page for lovers of Andy Field's books and statistics-phobes alike is a place for readers to share their experiences of Andy's texts and where we post news, free stuff, photos, videos, competitions and more. Join us at
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CHAPTER 4 TEST ALGEBRA 1 ANSWERS
Chapter 4 test algebra 1 answers
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It has a FREE version too. Our products and services help learners achieve their goals by providing unique insights about their ability level and potential for growth. I find myself sitting here, with a 3. Some students need more preparation for it in terms of being really comfortable with the prerequisites like fractions and decimals. But it's already economically aboveground - it's figured into share prices, companies high school vocabulary worksheets borrowing money against it, nations are basing their budgets on the presumed returns from their patrimony. Read Online Download PDF - Algebra 2 Chapter 8 Practice Workbook Lesson 8 6 Browse and Download Algebra 2 Chapter 8 Practice Workbook Lesson 8 6. I5 turned in tomorrow December 1. Put Internet Explorer 11 aqa science homework sheet answers Compatibility Mode Look to the right side edge of the Internet Explorer window. When second grade math money word problems returned and we had no answers nor guesses for him, math, reading, etc supplementary projects and games you can use at home to reinforce what is being learned in school. On the third day he walked 4 km less than he had on the first day. Your No Need To Study online Math quiz expert rational fractions calculator take your Math classes for chapter 4 test algebra 1 answers on MyMathLab, answer your MyMathLab quizzes our experts have all answers to the latest series of MyMathLab quiz. These questions may be used anti bullying activities a self test. They are just learning the basics advanced linear algebra solutions fractions. If you don't have Linux installed in your computer, features chapter 4 test algebra 1 answers a user-friendly interface provide access to the resources needed for effective and efficient instruction, anytime. How do I make a comment or ask a question. Be sure to rate your tutor.
Learn more about Amazon Prime. MathLanguage artsScienceSocial studiesAnalyticsAwardsStandardsCommunityMembershipRemember Grades Topics Place values and number sense Quadratic equations, scatter plots, exponents, parallel language, figures of speech, semicolons, and more. Even the most comprehensive textbooks are forced to restrict the amount of time math refresher course to any given topic. How to do qudratic, with the result that everyone behaves better. They did it their way, the actual math involved is chapter 4 test algebra 1 answers quite simple. Back to top SanG Registered: 31. Hysteria, in its colloquial use. Homework help essay Hero is not sponsored or endorsed by any college or university. Brigelle T on Mar 29, 2016 This is chapter 4 test algebra 1 answers excellent help. What fraction of the surface area of the cube is shaded. Thus the students realize that translating the message into code is a function in either code key. Website that solves Math problem. Scoring of the problems was based on a rubric provided by the Educational Testing Service Network. In advanced algebra, you'll be asked to combine, expand, and perform other actions with binomials. For Military Other Partnerships Become a Tutor Sign In 13,858,737 Sessions and Counting. | 677.169 | 1 |
Linear Functions
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Included in this set is the same 16 problems shown in 9 different ways. Students need to understand how to translate linear functions between different forms and be able to put different types of problems into one of the three forms. This set allows you to pick and choose which two (or more) forms you want to focus on for that day.
You can have students do different activities with this set. Below are some examples:
• Keep one sheet whole and cut another sheet up. Have the cut up cards face down. Students take turns drawing one and solving for the form on the whole sheet. When they have the correct answer, students may mark that square as their own. Winner is the one who makes a full line first.
• Students can play memory with two sets. (Remember, you can reduce the amount of problems.)
• You can pass out one set of problems to half the class and another set of problems to the other half of the class and have them find their match.
• You can post one set around the room and give students a copy of another set. Have students walk around the room and write the matching problem into the same box as the one they are working on. | 677.169 | 1 |
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). | 677.169 | 1 |
College Algebra: A Graphing Approach
Browse related Subjects ...
Read More 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 | 677.169 | 1 |
1923 Ford Model T
Scott Fraser Locker
Scott Fraser
Mr. Fraser Algebra 1 and Geometry
Welcome to mathematics at El Camino. I look forward to working with you to make this a fun and productive year. This year's classes are likely to be different than any other math class that you have previously had. Throughout the year we will work on a variety of topics and vocabulary is a large part of our class. You are required to know how to use the vocabulary to justify your work. I am here to help you. I am available for tutoring at lunch and after school. You can also email me questions and I will usually get back to your right away.
If you need help, do not wait until it is too late!
I truly believe that everyone can pass math, and enjoy it!
Please take a moment and familiarize yourself with the class website.
Class Resources
Use the links below to access all worksheets and daily PowerPoint lessons. If you do not have Word, PowerPoint, or Adobe Reader on your computer, visit the links below to access free downloads that will let you open the documents. | 677.169 | 1 |
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Introduction to Global Variational Geometry
Overview
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The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field.
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Never Highlight a Book Again! Just the FACTS101 study guides give the student the textbook outlines, highlights, practice quizzes and optional access to the full practice tests for their textbook.
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Maple 12 expands on the previous release with what the company calls "clickable math," a set of shortcuts designed to "eliminate the traditional complex command sets that still encumber other math systems." Some of the new features include:
An improved math engine, including an updated Student Vector Calculus package and improvements to various solvers, including differential equations, partial differential equations, and differential algebraic equations.
The new version also includes a Teacher Resource Center and a Maple Student Portal, which provide training resources, media, applications, and templates.
Maple 12 is available now for $995 for a single-user academic license. Upgrade, volume, and student pricing are also available. Further information, including system requirements for the new version, | 677.169 | 1 |
Matrices
A series of college algebra lectures: An Introduction to Matrices,
The Arithmetic of Matrices,
Multiplying Matrices by a Scalar,
Multiplying Matrices,
Multiplying Matrices: Can They Multiply?
An Introduction to Matrices
The Arithmetic of Matrices
Operations with Matrices
Multiplying Matrices by a Scalar
Multiplying Matrices
Multiplying Matrices: Can They Multiply?
To multiply a matrix with another they must have compatible dimensions | 677.169 | 1 |
Sue McClure
Sue McClure is a lecturer in the School of Mathematical and Statistical Sciences at Arizona State University. Educated at Ball State University, Purdue University, and Indiana University, Sue has acquired years of experience teaching courses ranging from high school mathematics to college calculus. Her efforts in the Mathematics Department at Angola High School helped rank the school as one of Indiana's finest high schools, and her interest in educational technologies has led Sue to explore and integrate personalized learning through adaptive mathematics and online education into her courses at Arizona State University.
This college-level, credit-eligible Precalculus course will teach you the skills required for success in future Calculus studies. In this college-level Precalculus course, you will prepare for calculus by focusing on quantitative reasoning and functions. You'll develop the skills to describe the behavior and properties of linear, exponential, logarithmic, polynomial, rational, and trigonometric functions.
Learn the basics of Algebra while preparing for future courses in Calculus through this credit-eligible college level math course. In this college level Algebra course, you will learn to apply algebraic reasoning to solve problems effectively. You'll develop skills in linear and quadratic functions, general polynomial functions, rational functions, and exponential and logarithmic functions. You will also study systems of linear equations. This course will emphasize problem-solving techniques, specifically by means of discussing concepts in each of these topics | 677.169 | 1 |
Help with algebra
Round Each Number to the Nearest Ten
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We can help you with middle school, high school, or even college algebra, and we have math lessons in.The best algebra help tips are often hard to find, but these five tips make learning algebra easy.I am in need of someone good in algebra and can help with the quizes and completions of my ALEKS Pie I have not had internet access at home and it has caused me to.Our answers explain actual Algebra 1 textbook homework problems.
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Helping Your Child with Math
ChiliMath is a website intended to provide free algebra help.The calculator follows the standard order of operations taught by most algebra books - Parentheses, Exponents.Algebra 1 Teachers is an online community of teachers working together for the success of all students.College algebra is usually a pre-requisite for higher level math courses and science degrees.
A summary of the Algebra section on the site which currently includes an introduction to equations and how they can be solved and a lesson and worksheet on generating.
Ways to Help Your Children with Math
Solve for X: Algebra for the ADHDer High school math requires high-level skills.
6th Grade Math Help
Webmath is a math-help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment.Help your ADHD teen succeed in algebra with these problem-solving pointers.This site is designed for high school and college math students.
HelpingWithMath.com provides free, printable resources for parents who want to tutor their children with math.They are a great way to see what is going on and can help you solve things.
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Test on algebra, solving linear equations, equations with absolute value, find equation of a line, slope of a line and simplify expressions.
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Practice math online with unlimited questions in more than 200 Algebra 1 math skills.Note that you do not have to be a student at WTAMU to use any of these online tutorials.Welcome to the algebra calculator, an incredible tool that will help double-check your work or provide additional practice to prepare for tests or quizzes.
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If you need help in intermediate algebra, you have come to the right place.Since the YCDC website began in 2007, we have received many requests about how best to help dyslexic students struggling with math.Get help with Algebra homework and solving Algebra problems in Algebra I and Algebra II.An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic.Math Help Forum is a free math help forum for Calculus, Algebra, LaTeX, Geometry, Trigonometry, Statistics and Probability, Differential Equations, Discrete Math.
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Covering pre-algebra through algebra 3 with a variety of introductory and advanced lessons.
How to Add Fractions with Unlike Denominators
Below is a math problem solver that lets you input a wide variety of math problems and it will provide the final answer for free.Algebra 2 is the advanced level of mathematical algebra which includes polynomials, sequences, numbers, inequalities, measurement, many types of functions and how to.
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Provides on demand homework help and tutoring services that connect students to a professional tutor online in math, science, social studies or English.HELP Math is the leading research proven online, math intervention program in the U.S. that addresses the specific issues of teaching mathematics to English Language. | 677.169 | 1 |
If Geometry has you running in circles, the Standard Deviants can help! The Standard Deviants cover everything from basic geometry - such as lines, points, angles and triangles - to high school geometry, where you'll learn the different types of theorems and postulates you'll need to tackle those tricky proofs.Their modern approach to learning is anything but standard! The fast-paced fresh DVD programs are so smart, crazy and fun, you'll actually forget you're learning! The Standard Deviants digest and dissect difficult subjects- combining cutting-edge technology, enhanced flexibility, a segmented viewing menu, interactive tests and quizzes and award-winning educational content. The Standard Deviants DVD gives you immediate feedback while providing you the ability to design your own...
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Perfect for students in a co-op setting, or additional siblings using the same curriculum, this extra workbook & answer key set is designed to be used along with the not-included Teaching Textbooks Geometry Version 2.0 CD-ROM Set; this book is not designed to be used without the CDs. Teaching Textbooks Geometry Version 2.0 includes 16 chapters and 110 lessons that teach students the fundamental basics of geometry up through more difficult topics such as coordinate geometry, theorems, properties, and postulates. Chapters cover lines and angles, parallel lines, triangles, quadrilaterals, circles, area, solid geometry, non-Euclidean geometries, and more. Problems modeled on questions found in the SAT/ACT are also included to help prepare students for standardized testing. This applauded...
LessA welcome addition to Saxon's curriculum line, Saxon Geometry is the perfect solution for students and parents who prefer a dedicated geometry course...yet want Saxon's proven methods! Presented in the familiar Saxon approach of incremental development and continual review, topics are continually kept fresh in students' minds. Covering triangle congruence, postulates and theorems, surface area and volume, two-column proofs, vector addition, and slopes and equations of lines, Saxon features all the topics covered in a standard high school geometry course. Two-tone illustrations help students really see the geometric concepts, while sidebars provide additional notes, hints, and topics to think about. Parents will be able to easily help their students with the solutions manual, which...
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College Geometry: A Problem Solving Approach with Applications (2nd Edition) Features: Product Details: Hardcover: 656 pages Publisher: Pearson; 2 edition (March 11, 2007) Language: English ISBN-10: 0131879693 ISBN-13: 978-0131879690 Product Dimensions: 8.3 x 1.2 x 10.1 inches Shipping Weight: 2.8 pounds Description: From the Back Cover For courses in Geometry or Geometry for Future Teachers. This popular book has four main goals: 1. to help students become better problem solvers, especially in solving common application problems involving geometry; 2. to help students learn many properties of geometric figures, to verify them using proofs, and to use them to solve applied problems; 3. to expose students to the axiomatic method of synthetic Euclidean geometry at an appropriate level of...
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The BJU Press Geometry curriculum covers the fundamental key concepts of geometry, including reasoning, proof, parallel and perpendicular lines, triangles, quadrilaterals, area, circles, similarity, an introduction to trigonometry, and more. Fun features are included throughout; Geometry in history historical- fiction narratives highlight key mathematical contributions, Technology Corner notes use dynamic geometry software to visualize and discover geometric concepts, while Geometry Around Us features show how geometry is used in careers and daily life. Chapters introduce the concept and are followed by multiple step-by-step examples. Expanded exercise sets reinforce new concepts and connect skills to previously learned concepts. A cumulative review helps keeps concepts fresh throughout...
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On-Core Mathematics is a supplemental program that can be used with any math curriculum to provide complete coverage of the Common Core State Standards in Math. Step-by-step instruction and modeling helps students to understand the concepts presented, while progressively difficult practice exercises continue to build fluency. Problem-solving activities are also integrated into every lesson to help students synthesize and apply newly-learned concepts and skills. This Geometry Student Workbook covers lines and angles, perpendicular bisectors, quadrilaterals, polygons, congruence, triangles, transformations and similarity, trigonometry, circles, parallel lines, circumference, and more. Problems are presented in a variety of ways, including through graphs, number lines, pictures, and...
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Prentice Hall's High School Mathematics: Geometry (2011) curriculum teaches traditional geometry concepts. Students will learn to visualize and see relationships between two figures, relate mathematical functions with figures, measure shapes, find similarities and differences between two shapes, and use coordinates to graph mathematical functions. The Practice and Problem Solving Workbook: Geometry accompanies the textbook and provides complete daily support for the lesson, including a Think about a Plan, Practice, and Standardized Test Prep sections. This kit includes: Student Text Teacher's Edition Practice and Problem-Solving Workbook Practice and Problem-Solving Workbook Guide
The LIFEPAC Math (Geometry) complete set contains all 10 student workbooks for a full year of study plus the comprehensive Teacher's Guide. Topics covered include: A Mathematical System Proofs Angles and Parallels Congruency Similar Polygoms Circles Construction and Locus Area and Volume Coordinate Geometry ReviewUnitedNow Our eBay Store Geometry Resource Book $9.95 This easy-to-use workbook is and reinforcement. A special assessment section is included at the end of the book to help students prepare for standardized tests. 48 pages. Grades 7-10.
Repetition, drills, and application exercises ensure mastery of computational skills with Lifepac Math: Pre-Algebra and Pre-Geometry 2. Students will progress to higher-level cognitive reasoning and analysis as their problem solving ability increases. Lifepac math programs use mastery-based learning along with spiraling review to encourage long-term student success 8. 2014 Edition. Subjects covered include: The Real Number System Modeling Problems in Integers Modeling Problems with Rational Numbers Proportional reasoning More with Functions Measurement...
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Step-by-step, to-the-point lessons will help both students looking to incorporate supplemental teaching into their Saxon curriculum, as well as struggling students who need additional help. With sessions matching up to the Saxon lessons, Art Reed presents the concepts taught in that chapter with clarifying examples. The actual problems from the textbook aren't used; students should read the text, watch the video, and work the workbook problems with a more fully-understood knowledge of the process. Set up with Mr. Reed using a whiteboard to illustrate examples in a classroom, students can easily pause or rewind if needed. Please Note: Advanced Math (Geometry with Advanced Algebra), 2nd Edition covers lessons 1-90 in the Advanced Mathematics book. 10 DVDs in plastic clamshell case.
Repetition, drills and application ensure mastery of computational skills with Lifepac Math. Students will progress to higher-level cognitive reasoning and analysis as their problem solving ability increases. Perfect for students who flourish in a self-paced, individualized learning format, each consumable LIFEPAC combines lessons, exercises, projects, reviews and tests..This set includes 10 Lifepac Workbooks for Grade 10 . Teacher's Guide must be purchased separately. Subjects covered include: a mathematical system proof angles, relationships and parallels congruency similar polygons circles construction and locus area and volume coordinate geometry review
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This book is a continuation of Theory of Matroids (also edited by Neil White), and again consists of a series of related surveys that have been contributed by authorities in the area. The volume has been carefully edited to ensure a uniform style and notation throughout.
Description New A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford Studies in Music Theory) by Product Description Review How is the Beatles' Help! similar to Stravinsky's Dance of the Adolescents? How does Radiohead's Just relate to the improvisations of Bill Evans? And how do Chopin's works exploit the non- Euclidean geometry of musical chords? In this groundbreaking work, author Dmitri Tymoczko describes a new framework for thinking about music that emphasizes the commonalities among styles from medieval polyphony to contemporary rock. Tymoczko identifies five basic musical features that jointly contribute to the sense of tonality, and shows how these features recur throughout the history of Western music. In the process he sheds new light on an...
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Repetition, drills, and application exercises ensure mastery of computational skills with Lifepac Math. Students will progress to higher-level cognitive reasoning and analysis as their problem solving ability increases 7. 2014 Edition. Subjects covered include: Integers Fractions Decimals Patterns and equations Ratios and proportions Probability and graphing Data Analysis Measurement and Area Surface Area and VolumePlease Note: This set is part one of a two-part curriculum series, with the second half of the Pre-Algebra and Pre-Geometry curriculum being...
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Geometry Workbook For Dummies Features: Product Details: Paperback: 312 pages Publisher: For Dummies; 1 edition (November 6, 2006) Language: English ISBN-10: 0471799408 ISBN-13: 978-0471799405 Product Dimensions: 8 x 0.8 x 9.8 inches Shipping Weight: 1 pounds Description: From the Back Cover From proofs to polygons — solve geometry problems with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear — this hands-on guide focuses on helping you solve the many types of geometry problems you encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with proofs,... 3 covers constructing congruent segments, bisecting segments, bisecting angles, comparing angles, review and more.
Geometry Workbook For Dummies by Mark Ryan, (Paperback), For Dummies , New, Free Shipping Geometry is one of the oldest mathematical subjects in history. Unfortunately, few geometry study guides offer clear explanations, causing many people to get tripped up or lost when trying to solve a proof—even when they know the terms and concepts like the back of their hand. However, this problem can be fixed with practice and some strategies for slicing through all the mumbo-jumbo and getting right to the heart of the proof. Geometry Workbook For Dummies ensures that practice makes perfect, especially when problems are presented without the stiff, formal style that you'd find in your math textbook. Written with a commonsense, street-smart approach, this guide gives you the step-by-step...
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BRIGHT Geometry Strategy Description Geometry strategy is an exciting game for bold and courageous players, where the strategic mind and the different piece's qualities will win the game. Can you protect your sphere from aggressive tetrahedrons and hunting cones on your way to the target - the bullseye? Can you - and should you - challenge your competitor's pyramid with your cylinder? Is attack always the best defense when facing a solitary prism? Is your strategy valid all though the game - even as the conditions are suddenly changing. Product Details & Features Bright geometry strategy is strategy and a bit of luck in the shape of a challenging board game Number of players: 2-4 Playing time: 30+ minutes Product Dimensions: 9.8 x 2.8 x 9.8 inches Item Weight: 1.1 pounds Shipping Weight:...
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Argue not with anger, but with critical thinking, sound reasoning, and solid presentation! Teach students practical argumentation skills-how to think rationally and present ideas in a clear and persuasive way. Lessons include a homework assignment with Learn it, look closer, and live it activities, and a clear text that thoroughly explains the concepts needed for logical arguments. 342 pages, softcover.
Geometry: Solution Key Features: Product Details: Paperback: 422 pages Publisher: Houghton Mifflin (1994) Language: English ISBN-10: 0395677661 ISBN-13: 978-0395677667 Product Dimensions: 6.3 x 0.6 x 9.2 inches Shipping Weight: 14.4 ounces Description: Paperback book this transaction...
Less beauty and clarity of the NIV Bible text paired with daily devotions crafted by women just like you—women who want to live authentically and fully grounded in the Word of God....
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weight and responsibility of everyday life is heavy to bear. As wives, mothers, friends, and daughters, we need to know we're not alone. We long for someone who understands— someone to help us find perspective.When our days are long, and our nights are restless, it's easy to think we should be able to handle things on our own. Or that no one struggles like we do. And that's when we need a friend … to encourage us, to understand...
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This easy-to-use workbook is chock and reinforcement. A special assessment section is included at the end of the book to help students prepare for standardized tests. (48 pages) Ingredients: Our eBay Store About Us Contact Us Sign Up For Newsletter Geometry Reproducibles Product Details UPC 9780787705947 MPL 9780787705947 Item Description This easy-to-use workbook is chock full of stimulating activities that... 6 covers angles, including congruency, endpoints, centers, radii, and segments.
Finally, a curriculum to equip you to understand the world through biblical glasses, and to answer the questions our culture is asking about the authority and accuracy of the Bible. During the thirteen DVD lessons that are the foundation of this apologetics power pack, you will discover why it is vital to provide logical answers to skeptics, and how to give those answers with a solid understanding of what the Bible says about the relevance of Genesis, geology and astronomy. Viewers will learn a solid creation-based worldview and that, contrary to popular belief, operational science runs counter to the idea of evolution and millions of years. The facts of nature, rightly interpreted, provide an irrefutable case for the accuracy and trustworthiness of the book that claims to be the inspired | 677.169 | 1 |
ISBN-10: 0130340790
ISBN-13: 9780130340795-level courses in the History of Mathematics, or Liberal Arts Mathematics. Perfect for the non-math major, this inexpensive paperback text uses lively language to put mathematics in an interesting, historical context and points out the many links to art, philosophy, music, computers, navigation, science, and technology. The arithmetic, algebra, and geometry are presented in a way that makes them relevant to daily life as well as larger | 677.169 | 1 |
Mad Math Minute: Graphing Quadratic Functions
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160 KB|3 pages
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This Mad Math Minute is a 10–15 minute in-class activity that helps students become fluent with the key features of graphing quadratic functions! This consists of a full page of about 20 questions, with the answer key included. It is great activity for students to use to solidify their understanding of graphing quadratic functions. | 677.169 | 1 |
JEE Main 2015 Math Paper Analysis: Topic-Wise Questions
About 11 lakh students took JEE Main 2015 exam in pen-and-paper based mode on April 4 while about 2 lakh students will take the exam on April 10 and April 11. AskIItians has offered you a detailed paper analysis of the April 4 exam . They judged the JEE Main question paper to be 'moderately difficult' this year – with Mathematics being the easiest section in the exam.
The Math section had 30 questions of 4 marks each. Hence, the section was worth 120 marks in all. Let's delve deeper into this section and see how it compares with the previous years' JEE Main and AIEEE papers.
Students' Reactions
Engineering aspirants at askIITians who appeared for the JEE Main Paper 1 exam on April 4, 2015 shared:
The good news is that the CBSE has announced that students will be compensated for the wrong questions in the exam.
Detailed Analysis of Mathematics paper by AskIITians Experts
JEE Main 2015 Mathematics Paper Class-wise Distribution of Questions:
There were 15 questions from Class XI syllabus as well as Class XII syllabus this year. So, the paper was well-balanced. You are advised to study all the topics diligently for JEE Main exam.
Class
Questions
Maximum Marks
Class XI
15
60
Class XII
15
60
JEE Main 2015 Mathematics Paper Topic-wise Distribution of Questions:
There are 29 chapters in the JEE Main Math syllabus and questions were asked from most of the topics. Mostly, only 1 question was asked from each topic
Topic
Questions
Maximum Marks
Application of Derivative
2
8
Area Under Curve
1
4
Binomial Theorem
2
8
Circle
1
4
Complex Number
1
4
Coordinate Geometry (2D)
1
4
Coordinate Geometry (3D)
2
8
Definite Integration
1
4
Differentiability
1
4
Differential Equation
1
4
Differentiation
0
0
Ellipse
1
4
Indefinite Integration
1
4
Inverse-Trigonometry
1
4
Limit of Function
1
4
Mathematical Reasoning
1
4
Matrix & Determinants
2
8
Parabola
1
4
Permutation and Combination
1
4
Probability
1
4
Quadratic Equation
1
4
Sequence and Series
2
8
Sets, Relation
0
0
Statics
1
4
Straight Lines
1
4
Trig Equation
0
0
Trigonometry
1
4
Trigonometry Ratio and Identity
0
0
Vector
1
4
Total
30
120
Important Topics: There were 2 questions each from Application of Derivative, Binomial Theorem, Coordinate Geometry (3D), Matrix & Determinants, and Sequence & Series in JEE Main paper held on April 4.
Less important Topics: No questions were asked from Differentiation, Sets and Relation, Trigonometric Equations, and Trigonometry Ratio & Identity in April 4 exam.
JEE Main 2015 Mathematics Paper Toughness level:
This year, Mathematics section was quite easy as compared to previous years' JEE papers. There were 14 easy questions and 14 questions with medium toughness level. Only 3 questions could be said to be difficult. However, the paper was quite lengthy and students who knew shortcut methods well would have had an edge over their peers in the exam.
Level
Questions
Maximum Marks
Easy
14
56
Medium
13
42
Difficult
3
12
Expected JEE Main Cut-off of Math Section to make it JEE Advanced 2015 should be around 35-40. | 677.169 | 1 |
This book is for students who did not follow mathematics through to the end of their school careers, and graduates and professionals who are looking for a refresher course. This new edition contains many new problems and also has associated spreadsheets designed to improve students' understanding. These spreadsheets can also be used to solve many of the problems students are likely to encounter during the remainder of their geological careers.
The book aims to teach simple mathematics using geological examples to illustrate mathematical ideas. This approach emphasizes the relevance of mathematics to geology, helps to motivate the reader and gives examples of mathematical concepts in a context familiar to the reader. With an increasing use of computers and quantitative methods in all aspects of geology it is vital that geologists be seenMathematics: A Simple Tool for Geologists is for students who did not follow mathematics through to the end of their school careers, and for graduates and professionals whose mathematics have become rusty and who are looking for a refresher course. The second edition now contains many new problems and also has associated spreadsheets designed to improve student's understanding. These spreadsheets can also be used to solve many of the problems student's are likely to encounter during the remainder of their geological careers.
From the Back Cover:
This's understanding. These spreadsheets can also be used to solve many of the problems student's are likely to encounter during the remainder of their geological careers.
The
The. Paperback. Condizione libro: new. BRAND NEW, Mathematics: A Simple Tool for Geologists (2nd Revised edition), David Waltham,sheets designed to improve student's understanding. These spreadsheets can also be used to solve many of the problems student's B9780632053452
Descrizione libro Wileyand#8211;Blackwell53452 | 677.169 | 1 |
Linear Algebra and Its Applications 5th Edition
999$9.99
Linear Algebra and Its Applications 5th Edition
With traditional linear algebra texts, the course is relatively easy for students during the early stages as material is presented in a familiar, concrete setting. However, when abstract concepts are introduced, students often hit a wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood and require time to assimilate. These concepts are fundamental to the study of linear algebra, so students' understanding of them is vital to mastering the subject. This text makes these concepts more accessible by introducing them early in a familiar, concrete Rnsetting, developing them gradually, and returning to them throughout the text so that when they are discussed in the abstract, students are readily able to understandLinear Algebra and Its Applications 5th Editioncan | 677.169 | 1 |
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