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What covers in a math workshop? •Review past/current contents each week. •Improve study and time management skills as well as reduce math anxiety. •Develop academic support network linked to specific classes with other students. •Work collaboratively to critically analyze course contents to improve understanding of complex material with workshop facilitators. •Provide opportunity to become actively involved in the course material. •Discover study and test preparation strategies. Do I need to sign up? No. Students just show up during the time scheduled.
Does it count for extra credit? It depends on each instructor. Does it count for the DLA? It depends on each instructor. Some instructors might allow Math 80/81 workshop to be counted as DLAs.
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Matrices
Matrices
Matrices
In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions,
arranged in rows and columns. The individual items in a matrix are called its elements or entries. An
example of a matrix with 2 rows and 3 columns is
Matrices of the same size can be added or subtracted element by element. The rule for matrix
multiplication is more complicated, and two matrices can be multiplied only when the number of
columns in the first equals the number of rows in the second. A major application of matrices is to
represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For
example, the rotation of vectors in three dimensional space is a linear transformation.
If R is a rotation matrix and v is a column vector (a matrix with only one column) describing the
position of a point in space, the product Rv is a column vector describing the position of that point after
a rotation. The product of two matrices is a matrix that represents the composition of two linear
transformations. Another application of matrices is in the solution of a system of linear equations. If the
matrix is square, it is possible to deduce some of its properties by computing its determinant. For
example, a square matrix has an inverse if and only if its determinant is not zero. Eigenvalues and
eigenvectors provide insight into the geometry of linear transformations.
Know More About :- Triangles
Math.Edurite.com Page : 1/3
Matrices find applications in most scientific fields. In physics, matrices are used to study electrical
circuits, optics, and quantum mechanics. In computer graphics, matrices are used to project a 3-
dimensional image onto a 2-dimensional screen, and to create realistic-seeming motion. Matrix calculus
generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix
computations, a subject that is centuries old and is today an expanding area of research. Matrix
decomposition methods simplify computations, both theoretically and practically. Algorithms that are
tailored to the structure of particular matrix structures, e.g. sparse matrices and near-diagonal matrices,
expedite computations in finite element method and other computations. Infinite matrices occur in
planetary theory and in atomic theory. A simple example is the matrix representing the derivative
operator, which acts on the Taylor series of a function.
A matrix is a rectangular arrangement of mathematical expressions that can be simply numbers.[1]
Commonly the m components of the matrix are written in a rectangular arrangement in the form of a
column of m rows: An alternative notation uses large parentheses instead of box brackets. The
horizontal and vertical lines in a matrix are called rows and columns, respectively. The numbers in the
matrix are called its entries or its elements. To specify the size of a matrix, a matrix with m rows and n
columns is called an m-by-n matrix or m × n matrix, while m and n are called its dimensions. The
above is a 4-by-3 matrix.
A matrix with one row (a 1 × n matrix) is called a row vector, and a matrix with one column (an m × 1
matrix) is called a column vector. Any row or column of a matrix determines a row or column vector,
obtained by removing all other rows or columns respectively from the matrix. For example, the row
vector for the third row of the above matrix A is
When a row or column of a matrix is interpreted as a value, this refers to the corresponding row or
column vector. For instance one may say that two different rows of a matrix are equal, meaning they
determine the same row vector. In some cases the value of a row or column should be interpreted just as
a sequence of values (an element of Rn if entries are real numbers) rather than as a matrix, for instance
when saying that the rows of a matrix are equal to the corresponding columns of its transpose matrix.
Read More About :- Whole Numbers
Math.Edurite.com Page : 2/3
Thank You
Math.Edurite.Com
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Browse related Subjects
...Read More while thoroughly covering the material required in an introductory algorithms course. Popular puzzles are used to motivate students' interest and strengthen their skills in algorithmic problem solving. Other learning-enhancement features include chapter summaries, hints to the exercises, and a detailed solution manual.Read Less
New. 0201743957
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MAG401: Geometry (2011-2012)
Major Concepts/Content: This course is designed to develop and promote student reasoning and problem solving involving geometric concepts and properties. Topics of study will include deductive reasoning using points, lines, and planes; segments, angles and triangles; quadrilaterals; polygons; and three-dimensional figures. Algebraic concepts are integrated with the geometric concepts throughout the course. Applications to real life situations are prevalent throughout the course.
Major Instructional Activities: Instructional activities include teaching students to plan, organize, and complete various forms of proofs using deductive reasoning. This course involves inductive reasoning, extended projects, classroom presentations by students, open-ended investigations, and written justification by students of the solution to the problems. Cooperative learning techniques and appropriate technology should be utilized throughout the course. Students should have access to calculators at all times.
Major Evaluative Techniques: Many evaluative processes will be used to assess student's written and oral work. These include but are not limited to multiple-choice, short-answer, discussion, or open-ended questions; structured or open-ended interview; homework; projects; journals; essays; dramatization; and class presentations. Students will also be required to successfully complete written tests, which present problems with a range of difficulty based upon expectations for the course. Testing formats will include restricted time tests, take-home tests, oral tests and student produced tests. Assessment methods can be supplemented by student-produced analysis of problem situations, solutions to problems, reports on investigations, and journal entries. Students will be provided the opportunity to do chapter projects that capture the concepts and skills presented throughout the chapter unit that emphasizes real world situations
Course Objectives: Upon successful completion of Geometry, the student
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Mathematical Applications and Modelling is the second in the series of the yearbooks of the Association of Mathematics Educators in Singapore. The book is unique as it addresses a focused theme on mathematics education. The objective is to illustrate the diversity within the theme and present research that translates into classroom pedagogies.The book,... more...
As a result of the editors' collaborative teaching at Harvard in the late 1960s, they produced a ground-breaking work -- The Art Of Problem Posing -- which related problem posing strategies to the already popular activity of problem solving. It took the concept of problem posing and created strategies for engaging in that activity as a central theme... more...
The authors show that there are underlying mathematical reasons for why games and puzzles are challenging (and perhaps why they are so much fun). They also show that games and puzzles can serve as powerful models of computation?quite different from the usual models of automata and circuits?offering a new way of thinking about computation. The appendices... more...
Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other areas. The book gives an overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical source... more...
Covers percentages, probability, proportions, and more Get a grip on all types of word problems by applying them to real life Are you mystified by math word problems? This easy-to-understand guide shows you how to conquer these tricky questions with a step-by-step plan for finding the right solution each and every time, no matter the kind or level... more...
The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions,... more...
Presented in this monograph is the current state-of-the-art in the theory of convex structures. The notion of convexity covered here is considerably broader than the classic one; specifically, it is not restricted to the context of vector spaces. Classical concepts of order-convex sets (Birkhoff) and of geodesically convex sets (Menger) are directly... more...
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9780077460396
ISBN:
0077460391
Edition: Eighth Publisher: Mc-Graw- Hill
Summary: Be guided through every step of the fundamentals of statistics. It is a great introduction to statistics for college students who have a basic grasp of algebra. It covers all the main concepts effectively and provides a lot of opportunity for practical application. Students are taught problem solving using detailed instructions and examples. It also focuses on the different digital applications used in statistics suc...h as Excel, graphing calculators and MINITAB. It also complements an online course so students can receive more from their course and excellent feedback from the online platform. We offer many top quality used statistics textbooks for college students.[read more]
ISBN-13:9780077460396
ISBN:0077460391
Edition:Eighthth
Publisher:Mc-Graw- Hill
is your source for cheap Elementary Statistics A Step By Step Approach + CD (Elementary Statistics A Step By Step Approach ( International Edition)) rentals, or new and used copies for sale.
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Mathematics 1431 - Precalculus I
"The most powerful single idea in mathematics is the notion of a variable." – K. Dewdney
COD's
Precalculus I (Mathematics 1431) course is "jam-packed full" of
the algebraic skills necessary to be successful in calculus. This online version
of the course covers the same content as the classroom version, but adds the
freedom to complete assignments and quizzes from "anywhere"….
Put on your math-pajamas and exercise your brain!
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More About
This Textbook
Overview
This book takes users step by step through the concepts of merchandising math. It is organized so that the chapters parallel a career path in the merchandising industry. The book begins with coverage of fundamental math concepts used in merchandising and progresses through the forms and math skills needed to buy, price, and re-price merchandise. Next readers learn the basics of creating and analyzing six-month plans. The final section of the book introduces math and merchandising concepts that are typically used at the corporate level. For individuals pursuing a career in merchandising
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Since some students have not completed the Geometry course by 9 th or 10 th grade any geometry ... Secondary math departments were able to adopt new textbooks in the spring of 2007 ...
Students taking Geometry or Algebra II in eighth grade or via ATYP have beendicult to place at times ... Textbooks The primary textbooks used will be: Calkins, Keith G. Numbers ...
This course does not necessarily have to be completed in the 9th Grade, but this ... 2 Algebra I Geometry Algebra II Advanced Mathematics Calculus Algebra I online Geometry ...
In geometry, the knowledge students acquired in Grade 8 about the properties of two-dimensional shapes is ... ways of working in mathematics; previewing of textbooks; pre ...
Examining the Everyday Mathematics textbooks for K - 5 I did not find the ... I do this experiment with my tenth gradegeometry class and the students thoroughly ...
... provides comprehensive practice with the math benchmarks tested on the 10th grade FCAT. ... assignment is meant to review topics for the FCAT Strands: Measurement and Geometry ... School Math Crunch Time Packet.pdf
This course does not necessarily have to be completed in the 10th Grade, but this ... 2 Algebra I Geometry Algebra II Advanced Mathematics Calculus Algebra I online Geometry ...
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A Mathematical Primer for Social Statistics 1st edition
1412960800
9781412960809
Details about A Mathematical Primer for Social Statistics:
John Fox's A Mathematical Primer for Social Statistics covers many often ignored yet important topics in mathematics and mathematical statistics. This text provides readers with the foundation on which an understanding of applied statistics rests.Key Features· Covers matrices, linear algebra, and vector geometry· Discusses basic differential and integral calculus · Focuses on probability and statistical estimation· Develops by way of illustration the seminal statistical method of linear least-squares regression Intended AudienceThis book is ideal for advanced undergraduates, graduate students, and researchers in the social sciences who need to understand and use relatively advanced statistical methods but whose mathematical preparation for this work is insufficient.
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Product Description
Bring in some outside instruction, and ensure that your students are really getting their Saxon math lessons! Designed to meet the needs of homeschoolers, Teaching Tapes features instruction by a state-certified teacher who explains and demonstrates each concept, example, and practice problem. Perfect for students working at their own pace, Teaching Tape DVDs will help students gain a solid understanding of the material they're working on. Each DVD is approximately 2 hours long. These DVDs cannot be used without the Saxon textbooks.
I have nothing against Saxon Algebra 2, its the teaching tapes that come with the textbook. They are incomplete. The teacher on the DVD covers the lesson content and how to solve (step by step) the example and practice problems. However, the problem sets are not covered at all.
This is difficult because the answer key does not give step-by-step instructions on how to solve the problems, it only gives the answer.
We found that the Saxon Teacher products does review all of the example, practice and problem sets along with giving a brief overview of the lesson. It's more complete and less expensive than the teaching tapes.
So, I'd pass on the more expensive teaching tapes and get the Saxon Teacher DVDs instead. They are a terrific resource.
Ask Christianbook
|
Q: What is the "technology tape DVD" on Saxon Algebra 2? How does this differ from the D.I.V.E. CD ROM?
A:
Both "Teaching Tapes DVDs" and "DIVE CD-ROMs" teach the concepts presented in the Saxon Math Books. They cover the lessons, investigations, and lesson practice problems. They do not review solutions to the problem sets or tests. The primary difference between the two is "Teaching Tapes" features an actual classroom teacher, while "DIVE" is in a digital whiteboard format where you hear the teaching, but only see a cursor illustrating the problems. Also, "Teaching Tapes" can be played on a computer or DVD player, while "DIVE" can only be used in a computer
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The primary purpose of A. G. Van Osdol and Company is to provide microcomputer equipment, computer networking technologies, mathematics tutoring, and personal experience that will assist adult learners to succeed in post-secondary education and training. The secondary purpose of this organization is to enable learners to improve academic and applied technical skills using mathematics. Learning the written language of mathematics will improve the learner's ability to read, reason, and write abstract concepts, ideas, and detailed information, enabling the learner to demonstrate their skills during their program of study and in their future careers.
Course contents can change from school to school, discipline to discipline, region to region. An intermediate course in one profession is often the beginning course of another, and an advanced course in yet another. Regardless of the topics in a course, our tutors will keep the learner on task and on time with the material specified by the instructor. The objective is to communicate with the learner in a manner that he/she can understand, minimizing the learner's frustration and maximizing retention and recall.
We provide individualized tutorial services in Algebra (Beginning, Intermediate, College Level, and Professional Technical), Trigonometry (Academic and Professional Technical), Linear Algebra, Statistics (Introductory and Professional Technical), Calculus (Single Variable Only)
Some of the difficulties experienced by student Veterans in school...
There are many problems that may interfere with an adult learner's ability to learn and demonstrate math skills. This list is not complete but represents some of the more common difficulties experienced by U. S. Military Veterans, Disabled Veterans, and many non-traditional students in our colleges and universities. More often than not these problems are dismissed rather than investigated and dealt with correctly.
The mathematics tutors we retain are U. S. Military Veterans who have "been there, done that" in mathematics. They know and can relate to challenges faced by Veteran students. The list below represents some of the ways that you might have changed since you were in school before your service, or they might describe problems you or your mentors (VA VR&E Counselor, instructor, advisor) may have observed. If you can identify with any of these problems our Veteran Tutors can show you how to overcome them and succeed.
Lose focus in the classroom.
Have cognitive impairments.
Cannot understand the teacher.
Pain or discomfort from old injuries.
Feelings of anxiety in the classroom.
Had a traumatic brain injury (TBI).
Are re-taking the course.
Your instructor or counselor suggests a tutor.
The longer you sit, the less you can remember.
Affected by post traumatic stress disorder (PTSD).
Cannot remember significant portions of lectures.
Problems taking notes and keeping up with the lecture.
Cannot do a problem by yourself without explaining it to someone else.
Not able to obtain individualized tutorial assistance at your school.
Grades consistently a "C" or lower.
Have failed the course previously.
You have a disability you want kept private.
Our Math Tutors
Meet our Math tutors! Well, one math tutor anyway. As our client base develops we will seek out other qualified Veterans as math tutors and add their information here.
If you are one of the multitude of people in a post-secondary school who believes mathematics is a series of painful tests you must pass with a "C" or at least a "D" before you can graduate then you need more information. Mathematics in the world today represents higher order thinking skills and abstract reasoning. It is used to communicate how everything works. It is a language you can use to solve problems and communicate precise information to others. Without mathematics there would be no society. No business, no communications, no cellular phones, no refrigerators, no shopping, and no abundant sources of food. Without mathematics society today would be very much like the Dark Ages.
Novice engineers and scientists often think that the math on a job is done by computer. Often it is, after it is done by people. Programed calculations can be interrupted, entered incorrectly or in the wrong order. We operate the computers; the computers do not operate us. So who checks the computer's calculations? People do. We check the computer, make changes, and re-run the programed calculations. The process of debugging a design is similar to "Wash, rinse, and repeat." This is how design errors are discovered.
One example of an extreme design error is in the units of measurement for an O-ring manufacturing specification. This error changed the size slightly, altering how the O-ring would contract/expand in outside temperatures colder than 40 degrees Fahrenheit. 7 lives were lost on February 1st 1986 when the space shuttle Challenger exploded shortly after takeoff. NASA and the manufacturer, Morton Thiokol, chose to disregard the warnings of the engineers who found the vulnerability months before.
If you are a member of a design team you may never know what your design will be used for. If you do, as in the case of the Challenger O-ring, or you are checking one of your peers work, you will need to communicate your findings. Would the administration of Morton-Thiokol and NASA be able to understand what the engineers found? If so then how could they approve a launch when it was 18 degrees Fahrenheit, 22 degrees colder than the coldest operating temperature of the O-ring? Mathematics is what allowed the engineers to find the design flaw of the O-ring. Administrators not understanding the mathematics describing the flaw is what killed the Challenger crew.
When you learn mathematics you will know how to identify, communicate effectively, and correct errors that may result in disaster for someone in the future.
The eInstructor.Net Meeting / Training room was available for use until the spring of 2014 by any of the companies and organizations who maintain their offices in the ISU Business and Technology Research Center. The room was equipped with an Interactive White Board (IWB) and projector, computer, and sound system which could be used in a variety of ways. The IWB was used to present Internet content or content from a laptop computer equipped with a wireless network interface card (NIC). Anything written on the IWB with its pens or your own finger could be saved for future reference. You could use Skype or another provider to share the IWB with multiple remote viewers. The entire presentation, including attendees and your interaction with remote viewers could be recorded for later editing and publishing.
Meeting and Training room was located in Suite 107 of ISU's Business and Technology Research Center
This resource no longer exists, but it did identify the need to develop the eInstructor.Net Supplemental Learning Management System. It would not have been possible to create the meeting / training room without the support of people in the community. Material contributions have been made by ISU's Business and Technology Research Center, ISU's RISE (Research & Innovation In Science & Engineering), ISU's Dept. of Dental Hygiene, ISU Networking and Telecommunications, the US Dept. of Veterans Affairs, and Columbia Computer Consulting. Thank you for your support on the eInstructor.Net project.
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Algebra 1 curriculum - recommendations for home schooling
In a nutshell, I recommend for most homeschooling parents to use a textbook along with some video instruction.
Why a textbook? Because it is good for students to learn to use a regular algebra textbook at this stage of their studies. It helps prepare them for any further studies (whether college or vocational) where they need to learn on their own, reading a textbook.
Also, good textbooks include not only basic exercises but also challenging ones. If you decide to go with some online algebra curriculum or video instruction, a regular textbook can act as a reference and as an additional problem "bank" for those challenging problems. You can also use it to check that your student is really getting instruction in all the typical algebra 1 topics.
Why videos? Because those replace the component that is present in regular classroom: the teacher explaining concepts and ideas. Learning algebra from a textbook alone might be too difficult for some students. If the parent cannot explain the math, videos will help bridge the gap. In today's world, there exist MANY free websites with algebra videos that can be used. And, some companies provide videos tailored to a specific textbook.
In this article, I first explain some basic options for algebra 1 in a homeschool setting. Then, some textbooks are described in more detail. The article also lists free algebra video websites, algebra online curricula, and gives a link to algebra tutorial website list.
VideoText Algebra - both algebra 1 and algebra 2. This one does not have a textbook but printed course notes and exercises (the videos are the "textbook"). I would consider purchasing some used textbook for a handy reference and for additional problems, especially for applications and word problems. $529
You can probably find these used for a cheaper price. Or, if you buy the product new, you can probably resell it later.
Use a regular textbook, along with videos from various math websites that may not be exactly tailored to the book. This is probably one of the most affordable options, because there exist so many sites offering free math videos.
Use a regular textbook, along with an online math curriculum. Similar to the one above, but more expensive. However, you get more support and resources, such as online quizzes, worksheets, reporting, etc.
An algebra course from one of the popular homeschooling companies. The drawback is that the authors of these products may not be math professors or even math teachers, and so the mathematics is often on a more "shallow" and superficial level than in textbooks authored by mathematics professors & teachers. Also, the explanations given for concepts may also be lacking in depth, and concentrate only on how to do the calculations, and not on the "why", or on the connections between concepts. Of course this varies from product to product, but I have seen some that judging from the samples were really "shallow".
However, I'm not saying these can't be used. If your student does not need a strong math background for future studies, they can be fine. Some families may also prefer the "format", whether it be many workbooks, a computer-based curriculum, or some other format.
What about Saxon algebra 1? The problem with Saxon is not the content itself, but how the lessons are organized. Saxon mixes in the topics and does not have chapters on certain broad topics, like other algebra books. For example, one lesson is on range, mean, median, and mode. The next is on conjunctions, the next is on percents, and the next one on polynomials—and so on. I don't think that is the best way to learn. You can read more about my opinion on Saxon math here.
What about Teaching Textbooks? This program is generally recognized as being quite easy and not challenging. The word problems I have seen in their samples are definitely too easy, for any grade I've looked at. Therefore, you cannot expect it to teach problem solving very well. This is actually unfortunate, because many parents and kids like its format. It would be a good program for low-performing students because it is so easy.
This is verified for example by this review left for my other math site:
We loved this curriculum until we looked at the Prentice Hall Algebra I book that the local high school was using. Then we realized that Teaching Textbooks Algebra I is way behind grade level! My son completed TT Algebra I and now is going through the Prentice Hall Algebra I book, to fill in the gaps, which are huge. This is taking him another 4-5 months! I had trusted Cathy Duffy's reviews of TT, and found I was wrong not to check it out more.
However, the method of Teaching Textbook is great -- the kids enjoyed doing it on their own, with access to the CD's and textbook.
If your student is college bound and plans to take the SAT and/or enter public high school, I would not recommend this curriculum.
Cindy
What about Singapore New Elementary Mathematic (NEM)? This series of books covers grades 7-10, and is considered to be quite challenging in its problems. Also, it may not have enough explanations of concepts, and the sequence of topics does not follow the traditional American way, but prealgebra, algebra, geometry, and statistics topics are mixed. If you feel comfortable with this, it will be fine, but I would only consider it for "mathy" kids because of the difficulty of the problem sets. Singapore Math also offers other series for grades 7-10 that are easier than NEM; please see a comparison here. Well-Trained Mind forum has lots of discussions about using NEM.
A regular algebra 1 textbook
If you go with the option #2 or #3 listed above, the textbooks that I recommend are:
The choice between the top three might not be easy. In a nutshell, Jacob's book is lively, concentrates on concepts, well-admired by a lot of people for its entertaining style, and has very good and interesting exercise sets. However, it is also a bit "lite" in content.
Foerster's book is considered one of the best, if not the best, by some. It goes much deeper, perhaps too deep for some students. It has superior, detailed explanations, lots of basic exercises, and challenging word problems.
Prentice Hall book is a regular, colorful schoolbook that is comprehensive in the content covered, and has a free online video & other resources to go with it.
The boxes below describe these books in more detail.
Harold Jacobs Algebra 1
Jacobs' book is kind of a literary work in itself. The instruction for a new topic always starts out with an interesting example from history, a cartoon, or such like. This makes the text livelier and easier to read, and can make math "more fun" in a sense. The exercise sets also include some very interesting problems that tie in with history or are otherwise amusing or amazing! The interesting problems thrown in help build mathematical intuition AND the love for mathematics.
One of the main drawbacks though is that the author often gives minimal explanations in the text, and not many worked-out examples. This is because a lot of the learning is supposed to happen within the exercises, which often follow the "guided inquiry" method of instruction.
There are three exercise sets for each lesson, of which set 1 is always review. There are answers in the back of the book for each exercise set 2.
Jacobs' algebra is also on the easy side, as far as CONTENT goes. While it does have the same chapters as any regular algebra book, in several topics, it does not cover the same depth as Foerster's or other algebra books. I'll give you some examples.
In inequalities involving absolute value, Jacobs only teaches inequalities that have "x" without a coefficient, for example | x + 4 | < −5. He does not include inequalities of the type | 2x + 4 | < −5. Also, he does not cover inequalities with two variables at all (for example, y > x + 4). In radical equations, the problems are limited to such as have x under the radical sign, for example √x + 2 = 5. Problems that include both √x and x are not included (for example √x + 2 = 5 + x). Also, Pythagorean Theorem is not covered. Scientific notation is not covered. These lacks are not necessarily a problem, since any algebra 2 book will review all of algebra 1, and should cover those topics.
The book also seems to teach the math on an easier level than Foerster or the Prentice Hall book, often practicing visual models in detail, and using exercises that build student's conceptual understanding step by step. This can actually be of great benefit for students who are not ready for more "algebraic" or analytic reasoning. It also means Jacobs' book could easily be used with younger audiences—some proficient, "mathy" kids could even study it in 7th grade.
Ask Dr. Callahan sells a matching DVD for Jacob's book that has video lessons covering each topic and working examples as needed.
Foerster Algebra 1
With Foerster, you can definitely "hear" a teacher speaking to you through the text. The book is written to the student, with excellent, sometimes even pedantic, explanations. The text often includes little tips like what a classroom teacher might say, such as "the vinculum is supposed to extend over here" or illustrations why something cannot be done.
Foerster's algebra goes fairly deep into the usual topics of algebra 1. The book includes both basic and challenging exercises, including lots of word problems. It is an excellent textbook when it comes to explaining mathematics. Foerster's approach is very analytical and logical, relying on mathematical thinking—which can be very good for students who are going into sciences.
The main difficulty I can see for homeschooling parents is that some of the exercises are quite challenging. Thus, Foerster's book may be too difficult for some students.
Prentice Hall Algebra 1
To see this book for yourself, you can preview chapter 2 of the 2009 edition here. This is a basic, comprehensive textbook used in public schools. It is typical of modern textbooks in that it has quite a bit of color and multiple authors. The book is more "cut and dried", without the bits of humor that you find in both Jacobs' and Foerster's books. It seems to have sufficient amount of examples, and lots of varied exercises.
As a definite bonus, Prentice Hall Algebra 1 (2009) book has a free companion website. (If the link is broken, go to and navigate to the algebra 1 book.) You will find narrated, interactive video tutorials for every lesson. Additionally, the site has lesson quizzes, chapter tests, vocabulary quizzes, and some real-world applications to supplement Prentice Hall Algebra 1.
The following links go to Amazon.com. Hover over the links to see pricing.
The newer editions of Prentice Hall Algebra 1 (2011/2012 and 2015 editions) are fine products, also. These are aligned to the Common Core Standards, and thus emphasize conceptual understanding and mathematical modeling more than the previous editions. For example, the publisher has added "Error Analysis" exercises, "Reasoning" exercises, and rich mathematical and real-world problems to the book.
In fact, the newer editions sound better to me than the earlier ones; however there are two drawbacks: 1) They are also more expensive on Amazon—even used copies—and 2) The companion website for the new editions requires an access code. The access code is supposed to be simply the ISBN of the teacher edition, but I did not get it to work.
You can access an
online sample of Algebra 1 here. You will need to register first. If that link doesn't work, go to → mathematics → algebra, and you should see Prentice Hall Mathematics Algebra 1 listed.
This book is written by the founder of Art of Problem Solving, Richard Rusczyk. For each topic, there are many example problems with detailed solutions and explanations, through which algebraic techniques are taught. The explanations often highlight ideas on best problem solving approaches, which is something you don't usually see in regular algebra textbooks. Exercises for the student follow.
Introduction to Algebra book goes through all the typical algebra 1 topics, plus present several topics that are usually part of Algebra 2 curriculum, such as exponents and logarithms, quadratic inequalities, functions, and complex numbers. The solutions manual includes full solutions, not just answers, which is a definite plus.
This book is indeed quite good for its intended purpose. It contains challenging problems, and is especially meant for "high-performing" math students, because it emphasizes problem solving, proof, and challenging problems (please see features here). It is NOT for weak or average students or for those who do not like problem solving. Please check out the long excerpts (samples) on AOP website to see if the book would be a good fit for your student.
Free algebra video websites
You can use these FREE video websites to accompany any algebra textbook you might have. To use them, first check the topic of the lesson in your textbook. Then find matching videos on these sites. You can definitely also leave this task to your student: he/she can read the textbook text, find videos, and then watch one or several videos on the same topic, before attempting to do any of the exercises in the textbook.
MathTV.com
Over 6,000 free, online video lessons for basic math, algebra, trigonometry, and calculus. Videos also available in Spanish. Also includes online textbooks. See also my review.
An online algebra curriculum
There exist several commercial online curricula offering algebra courses to homeschoolers. I cannot really say which ones would be the best since it's not possible for me to review and evaluate all of them, so that will be left to you.
These curricula are always based on either videos or animated tutorials. The exercise sets may be less comprehensive than those found in regular algebra textbooks, and often focus on the most basic types of exercises. You usually get access to simple online quizzes and a system that tracks student progress. You may even get one-on-one support from a tutor or teacher.
These resources may be good for a computer-oriented student. Personally I would augment the online curricula with more challenging problems from some algebra textbook.
I Can Learn Online
Interactive, animated courses for fundamentals of math, prealgebra, and algebra. Subscription fee of $30/month gives you access to all three courses. A free trial available.
ThinkWell
Multimedia video lectures that take the place of a traditional textbook, plus automatically graded exercises & homework. Titles offered are from grade 6 through calculus. The teacher on the videos is Edward Burger, who has a unique and intuitive approach to learning math. Online access to any one course $125 a year.
Educator
A collection of lectures by college professors, including algebra, trigonometry, calculus, and statistics courses (for high school/college). Subscriptions $45 first month, $35 monthly thereafter, $240 a year; they give you access to all courses.
Conquermaths
An online maths tutoring system with 480 full audio/visual lessons presented by a real teacher, synchronised with animated graphics and backed up by tests and progress reports. For UK key stages 3 and 4 (11-16 year olds). 16 free trial lessons. Subscriptions £15.95 a month, £69for six months; £99 a year.
Maths Power
Online math learning system with animated lessons, worksheets, and access to teacher support for K-12. Created from the Australian syllabus. Software version 275 AUD/year; online version 214 AUD/year.
Mathematics.com.au
Animated maths lessons, worksheets, topic tests and worked solutions for years 7-12 in the Australian curriculum. 32 free lessons available to
trial online. Online membership $26.95 AUD month or $127 for six months; or $197 AUD year; family plans available. Also available on CDs.
Free online algebra tutorials
There exist dozens of algebra websites, which usually have short tutorials on algebra topics. Some of them have short quizzes also. They wouldn't replace a textbook because of lack of exercises, but sometimes it's helpful to read several explanations for the same concept.
Please see a LONG list of these algebra websites here. The list also includes algebra worksheet sites, online calculators, and a few algebra games.
Math Mammoth Tour
Confused about the different options? Take a 7-day virtual email tour around Math Mammoth! You'll receive:
An initial email to download your GIFT of over 350 free worksheets and sample pages from my books;6 other emails on 6 subsequent days that describe the different series of Math Mammoth products, answering the most commonly asked questions, including "What is the difference between all these different-colored series?"
This way, you'll have time to digest the information over one week, plus an opportunity to ask me personally which book would be right for your child or students.
Note: You will FIRST get an email that asks you to confirm your email address.
If you cannot find this confirmation email, please check your SPAM/JUNK folder.
"Mini" Math Teaching Course
This is a little "virtual" email course. You will receive:
An initial email to download your GIFT of over 350 free worksheets and sample pages from my books;7 articles on important topics on mathematics education, including: multiplication tables, fractions, how to help a student who is behind, and the value of mistakes;1 email about Math Mammoth books.
Note: You will FIRST get an email that asks you to confirm your email address.
If you cannot find this confirmation email, please check your SPAM/JUNK folder.
Join over 25,000 others!
Maria's Math News is my newsletter, filled with math teaching resources, Math Mammoth news & discounts, humor, giveaways, and more! The content is equally good for all of us who teach math. :) The newsletter comes out once or twice a month. Peek at the previous volumes here.
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Develop Algebraic Thinking 6-8 - MAT ONLINE course will help teachers discover how to support deeper understanding of foundational algebraic concepts in grades 6-8. Teachers will explore growth patterns and functions, variables, linear relationships, and coordinate graphs. Based on the included text and research-based journal articles, teachers will design and implement a unit of grade-level appropriate algebraic thinking activities with their students. All of the readings and activities are built upon the Common Core standards. Teachers may complete this course with or without students.
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"I can work on the coursework in my home. No commuting! Best of all the course work helps me fine tune my curriculum work. Our district, like many, does not have as many PDP opportunities for our core subjects. I find that the courses are interesting and priced well. Thanks Fresno Pacific!"
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Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also
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Rarely does there appear an undergraduate textbook that begs to be taught from even as it leaves its reader wanting to take the class from its authors. Peter Hilton, Derek Holton and Jean Pedersen have produced a gem in this genre. The authors have written a set of independent mathematical essays that explore topics and interrelationships for which there is rarely room in the undergraduate curriculum.
Mathematical Vistas: From a Room with Many Windows (MV) is a companion volume to the authors' 1997 work Mathematical Reflections: In a Room with Many Mirrors (MR). Because of the close interrelationships between MV and MR, this review discusses aspects of both texts. The authors' intended audience is readers having at least the mathematical maturity of bright secondary mathematics students (for MR), bright college or university freshmen, and "adults wishing to update and upgrade their mathematical competence". There is a clear invitation for teachers of mathematics to read these volumes, particularly chapter 9 of MR in which the authors have written an essay on how students best may learn mathematics and how teachers may apply their "principles of mathematical pedagogy".
This series is much more than it would seem since many descriptive terms can mislead one to expect less of these books than there is. The Library of Congress In-Publication Data categorizes it as "1. Mathematics — Popular works", but one would be very wrong to confuse this series with the rash of simplistic popularizations that one finds on today's marketplace. The authors state that these books are intended to convey "what mathematics is about, and how it is done". And so they are — but the about part includes the relevant details and proofs, while the done part concretely addresses the processes of synthesizing prior theorems and attempting to formulate and prove generalizations.
To be sure, there is a lot in these books that forces reader participation and that is, frankly, fun. Magic tricks based on the Fibonacci series, paper folding and polyhedra construction, and matching boys and girls at dances are examples the authors use to lead into deeper, and sometimes surprising, mathematical topics. The authors often speak directly to the reader. They point out areas where the reader may have become confused or where the reader needs to look for subtleties in the exposition. Problems and thought-provoking questions are interspersed as "Breaks" at appropriate points in the exposition. Clearly, many of these breaks are intended to lead the reader to anticipate the direction in which the text's investigation is about to turn.
Each book contains nine more or less independent chapters that can pretty much be read in arbitrary order. In MV, these concern:
mathematical (mostly statistical) paradoxes;
a history of elementary approaches to proving Fermat's Last Theorem with a commendable summary of Wiles' proof;
deriving properties of the Fibonacci and Lucasian numbers (as an extension to a comparable chapter in MR);
relationships between number theory, paper-folding and polyhedra-building (also an extension to a preliminary chapter in MR, but culminating in a proof of the General Quasi-Order Theorem);
explorations of graph theory and its use in proving the Four Color Theorem;
generalizing from binomial coefficients and the Pascal Triangle to multinomial coefficients, the Pascal Tetrahedron, and more as extensions to a comparable chapter in MR;
the Catalan numbers and their generalization;
symmetry, group theory and a motivated statement (without proof) of the Pólya Enumeration Theorem; and
Needless to say, there's a lot here. (In addition to three chapters cited above, MR also covers linear recursive definitions; modular arithmetic; quilting as a motivation for tessellation of the plane; large numbers through orders of infinity; and the mathematics of fractal geometry).
The level of treatment is appropriate to the undergraduate audience. Arithmetic and algebraic manipulations are treated in close detail, often so that a method of manipulation is pointed out that can be used as a toolbox technique for proving or deriving similar theorems. In many cases the exposition is similar to the way a teacher would perform at a blackboard, using cartoon bubbles that point to portions of an expression that have a special characteristic or that will be manipulated in a special way. Conceptually difficult passages are clearly marked as such for the reader. The authors devote a lot of space to instructions illustrating the manipulations of paper folding and polyhedron-building so that the student can produce and work with three-dimensional models. Such models can, I believe, prove to be particularly helpful in explaining the concepts behind the Pólya Enumeration Theorem. Relevant references are cited in each chapter. Sometimes, the extended chapters come close to becoming tedious as properties of a mathematical structure are examined through a suite of related manipulative derivation and proof techniques. But the reader can feel the enthusiasm of the authors for deriving new results and the joy of generalizing upon them. Such pedagogical enthusiasm is quite contagious!
The authors' personal feelings about "how mathematics should be done" are central to the text of both volumes. They are stated, with illustrations, in chapter 9 of MR, and are worthy of educators' attention. I believe it is worth listing a few of them here:
Mathematics is only done effectively if the experience is enjoyable.
Mathematics usually evolves out of communication between like-minded people; human beings can only communicate interesting ideas informally.
Never be pedantic; sometimes, but by no means always, be precise (the "Principle of Licensed Sloppiness").
Elementary arithmetic goes from question to answer; but genuine mathematics also, and importantly, goes from answer to question.
Algorithms are first resorts for machines, but last resorts for human beings.
Use particular but not special cases.
Symmetry is a pervasive idea in mathematics; always look for symmetry.
Use appropriate notation and make it as simple as possible.
Mathematics should be taught so that students have a chance of comprehending how and why mathematics is done by those who do it successfully ("Basic Principle of Mathematical Instruction").
In some developments, the authors lead the reader to a point where it appears that a lot of tedious algebraic expansion, collection of terms, and other manipulation will be required. With sparkling freshness, they instead let the reader know that they don't enjoy "slogging" through a proof, particularly if mechanical slogging doesn't lead to achieving understanding. And so, appealing to a principle of being optimistic, they show the reader a variety of useful algebraic techniques that lead to the construction of conceptual proofs. The applications of chapter 9 make MR and MV a joy to read, and I'm certain that they will prove to be a delight to teach from!
There are only a few unfortunate typos in MV or remaining in the 2nd [Corrected!] printing of MR.
There is some duplicated text between the three pairs of corresponding MR and MV chapters mentioned above. But without question, the continuations in MV provide genuine extensions to the foundations laid in MR. Although it is possible to read MV without first having MR in hand, it is this reviewer's opinion that to do so would not be easy for the targeted audience — and it would also deprive the reader of a tremendous number of mathematical treats and insights. The writing is seamless, thought-provoking and entertaining. You and your students deserve to read both books!
Marvin Schaefer (bwapast@erols.com) is a computer security expert and was chief scientist at the National Computer Security Center at the NSA, and at Arca Systems. He has been a member of the MAA for 39 years and now operates an antiquarian book store called Books With a Past.
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Find a Sharpsburg, GA Algebra 2You will increase and improve computation of mean, mode, median, reciprocals, proportions, and factorials. Learn how to factor and expand math equations containing unknowns and coordinates on the x,y axis grid. Reading instruction is more than word calling
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An illustration of the basic structure of problem solving. The algebraic operations
of substitution and simplification are done error free using SureMath - the math software for 21st century problem
solving.
The starting equation RESPONDS directly to what is ASKED FOR and
REQUESTS additional information.
The next step RESPONDS to the request.
Supplying the RESPONSE to the requesting equation produces a RESULT.
The REQUEST-RESPONSE-RESULT structure will solve any problem.
The RESPONSE is shown indented, which indicates that the response is a subproblem of the main problem. This is the same type of logical organization that is familiar in forming a table of contents, an index, an organizational chart, computer programming and many other common methods of presenting information.
An organized presentation of a problem solution enhances communication between the problem solver and himself as well as with users of the solution.
EQUATIONS TALKThis problem solution was developed using SureMath, the problem solving software for the 21st century.
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Oaklyn Precalculus introduces the student to analytic geometry and the critical notions of functions, limits, and derivatives. Many types of functions are studied; linear, quadratic, exponential, trigonometric, and others. Additional topics include vectors, polar coordinates, parametric equations, conic sections, and more
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The National Council of Teachers of Mathematics' Principles and Standards for School Mathematics (2000) identifies algebra as a strand for grades Pre-K-12. The Standards identify the following concepts that all students should cover and comprehend:
Note 4
•
Understand patterns, relationships, and functions
•
Represent and analyze mathematical situations and structures using algebraic symbols
•
Use mathematical models to represent and understand quantitative relationships
•
Analyze change in various contexts
For the classroom in grades 6-8, understanding patterns includes the following expectations:
Relate and compare different forms of representation for relationships
•
Model and solve contextualized problems using various representations, such as graphs, tables, and equations
In this part, we'll look at problems that foster algebraic thinking as it relates to these standards, and explore ways of asking questions that elicit algebraic thinking. The situations we'll be exploring are representative of the kinds of problems you would find in some existing texts; in fact, you may recognize some of them! The goal is for you to examine these problems with the critical eye of someone who has taken this course and is beginning to view algebraic thinking with a different perspective.
Consider the situation below, appropriate for exploration in a grade 6-8 classroom:
Tat Ming is designing square swimming pools. Each pool has a square center that is the area of the water. Tat Ming uses blue tiles to represent the water. Around each pool there is a border of white tiles. Here are pictures of the three smallest square pools that he can design, with blue tiles for the interior and white tiles for the border. Note 4
What questions would you, as a mathematics learner, want to ask about this situation?
Problem B2
How do your questions reflect the algebra content in the situation?
Now focus on the questions you want the students in your classroom to consider. You may want to consider new ways to represent the relationships between the number of tiles of each color and the number of the square pools, and then use those representations to predict what will happen when the pools are very large.
Problem B3
What patterns, conjectures, and questions will your students find as they work with this situation?
Problem B4
What questions could you as the teacher pose to elicit and extend student thinking at your grade level?
Problem B5
Recall the framework you explored in Session 2 in looking at patterns: finding, describing, explaining, and using patterns to predict. Which of these skills will your students use in approaching this problem?
Problem B6
Read the article "Experiences with Patterning" from Teaching Children Mathematics. What ideas mentioned seem appropriate for your classroom? Note 5
Problem B7
In this sequence there are 4 toothpicks in Term 1, 7 toothpicks in Term 2, and 10 toothpicks in Term 3.
How many toothpicks are in Term 4? If you continued the pattern, how many toothpicks would you need to make Term 5? Term 6? Term 10?
What questions could you ask to develop students' skills in describing this pattern?
Problem B8
What questions could you ask to develop students' skills in predicting?
The swimming pool problem adapted from Algebra in the K-12 Curriculum: Dilemmas and Possibilities, Final Report to the Board of Directors, by the NCTM Algebra Working Group (East Lansing, Mich.: Michigan State University, 1995).
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Introduces the student to discrete structures and mathematical tools which are used to represent, analyze, and manipulate discrete objects. These include sets, functions, relations, graphs, combinatorics, discrete probability, recurrence relations, mathematical induction, symbolic logic, and graphs and trees.
Detailed Description of Course
Course content includes:
Combinatorics
Critical Path Analysis
Matching Problems
Algorithms for Solving Problems
Permutations and Combinations
Basic Probability
Pigeonhole Principle
Multiplication Principle
Addition Principle
Sets
Set Operations
Equivalence relations
Division Algorithm
Congruence
Functions
Logic
Mathematical Induction
Symbolic Logic and truth tables
Methods of Proof
Graphs
Representations
Di-graphs
Paths and Circuits (including but not limited to Euler and Hamiltonian circuits and paths)
Trees
Spanning Trees
Minimal/Maximal Spanning Trees
Depth-First Search
Binary Trees
Recurrence relations
Sequences
Linear Functions
Pascal's Triangle
Binomial Theorem
Detailed Description of Conduct of Course
In addition to lecture, students will work collaboratively on assignments created to help students understand the application of discrete mathematics concepts used in problem solving. Calculators and computers will be used to present and work the material in and outside class.
Student Goals and Objectives of the Course
In accordance with the NCATE standards for discrete mathematics, students will be able to demonstrate knowledge of the concepts of discrete mathematics such as (but not limited to): Perform operations on sets, prove logical statements using truth tables, prove problems by mathematical induction, use counting properties to solve combinatorics problems, understand basic principles of Graph Theory such as: path, cycle, connected graphs, subgraphs, etc., determine the shortest path in weighted graphs as it occurs in practical problems, and understand and apply trees and (minimal) spanning trees. To problem solve discrete mathematics problems; students will understand the application of an algorithm by applying them to problem situations such as those involving search and optimization. Students will develop the ability to communicate mathematically.
Assessment Measures
Students will demonstrate content understanding via written (and/or oral) exams, written homework problems, collaborative work in class, and class discussion. Students may be required to complete a project.
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Projects for Calculus is designed to add depth and meaning to any calculus course. The fifty-two projects presented in this text offer the opportunity to expand the use and understanding of mathematics. The wide range of topics will appeal to both instructors and students. Shorter, less demanding projects can be managed by the independent learner, while more involved, in-depth projects may be used for group learning. Each task draws on special mathematical topics and applications from subjects including medicine, engineering, economics, ecology, physics, and 18, 2002
wish I could give it a lower rating...
If you can understand anything in this book you probably don't need this book. I taught my third math class ever from this text and it's not worth the paper it's printed on or the time Stroyan invested to write it. Poor examples, ambiguous explanations, and some terms I'm pretty sure Stroyan just made up. It seems to substitute 'pretty pictures' for real math but expects you to understand both. Students will find this text severely frustrating and teachers will find that they have to teach twice as hard just so a few people can understand. Avoid this book or any classes that use it.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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2D geometry is basic to so many things, including any attempt at programming graphics. This book aims to introduce the basic ideas without frightening the non-mathematical reader.
It sort of works - but there are some reservations.
Following the usual form of a "Head First" title there are lots of photos, asides, quizzes, activities and so on - and it does help.
The ideas are all introduced as part of a "real world" problem - a fictitious homicide is used to introduce angles and properties of angles as you try to work out the CSI-take on geometry.
Later stories are similarly angled (pun intended) to capture the imagination of teenagers - skate boarding, designing a pattern for a screen graphic and so on. Again all of this mostly works even if you initially might feel a bit embarrassed by the cartoon-like characters - hey dude where's my angle. But if you stay with it you do get immersed in the problems and the geometry needed to solve them.
The book really does start from the basic idea of what an angle is and doesn't get very far. You do learn Pythagoras via the usual geometric demonstration rather than by proof and both proof and algebra are down played at every turn.
It is about getting the student to really understand and imagine the geometric properties under discussion. This is geometry by feel and experience, rather than proof, and there is nothing wrong with the approach.
This is not a book that is going to be of any use to the student who has even the slightest grasp of, or aptitude for, math and geometry in particular. It is such a low level and such a low information density that it really is only for the maths non-starter or refuser.
So as long as you realise that this is very basic 2D geometry and it isn't going to be of direct help if you are struggling with a traditional course on geometric proofs then it's a good book.
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Quantitative and Analytical Reasoning Courses
Formal reasoning and the formality of logic are central tools for decision-making in an uncertain world, and we expect our graduates to be conversant with mathematics and quantitative reasoning and to be able to apply quantitative reasoning to understand and solve everyday problems. Two quantitative courses are required for graduation (unless you successfully complete MATH 249). At least one of these credits will be intended to expand students' quantitative boundaries by providing the skills necessary to interpret and apply mathematics. Such courses are designated by the symbol (QA*) in the course schedule. The other course may be another (QA*) course or could be a discipline-based application of quantitative methodology, such as physics or computer science and is designated (QA). A list of fall courses that satisfy the quantitative requirement can be found in the 'Course Types' section of the Course Schedule.
Consult your Degree Audit to determine if you have fulfilled one or both of the quantitative requirements. If you have not already satisfied the quantitative requirement, you may wish to take one of your quantitative courses during your first semester. This is particularly recommended if you think you will go on to major in a field of study which requires a quantitative background.
Mathematics Placement
If you have not completed the quantitative requirement and elect to take a quantitative course in your first semester you will need to choose the right course for you. Based on your previous coursework and experience in mathematics, you can determine which of the following initial mathematics/quantitative courses would be most appropriate. The quantitative courses listed below have the QA* designation. Read the description of these courses carefully, mindful of your prior math preparation, and choose the level that matches your interests and abilities. A list of specific quantitative requirements for various majors is given below.
NOTE: Options 1 – 3 (below) have no particular prerequisites beyond algebra 2. Any of these courses would be appropriate for students who are primarily seeking to obtain a broad background and to fulfill the quantitative requirement. Students desiring a more technical quantitative background, particularly for use in mathematics or quantitative science, should plan to take courses in the calculus sequence (Calculus 1, Calculus 2, and Multivariable Calculus, options 3 – 5 below). Completing multivariable calculus MATH 249 successfully will satisfy both quantitative requirements.
If you opt to enter the calculus sequence, where should you begin? Advice for placement within the calculus sequence is provided below. You may also contact a member of the Mathematics Department for advice.
Contemporary Mathematics (MATH 130) - A survey of contemporary topics in mathematics such as: voting systems and power, apportionment, fair division of divisible and indivisible assets, efficient distribution, scheduling and routing, growth and decay in nature and economics, symmetry and fractal geometry, probability and statistics. This is NOT a remedial course. Prerequisite: two years of high school algebra.
Statistics (MATH 138) - An introduction to descriptive and inferential statistics. Emphasizes everyday applications and practical skills. This course is particularly recommended for students who neither need nor desire a calculus background, and is an excellent preparation for dealing with the statistics one encounters every day in our society. Prerequisite: two years of high school algebra.
The Calculus sequence
Calculus 1 (Math 141) – A first course in calculus which serves two purposes: 1 – a survey of calculus for those who want to know what calculus is "all about," but don't intend to take any more mathematics, or 2 – the first course in calculus which prepares the student to continue to Calculus 2. Topics include the differential and integral calculus of algebraic and exponential functions, together with their applications. Students who have taken a full year of high school calculus should begin calculus study with MATH 142 or MATH 249.
Multivariable Calculus (MATH 249) - Calculus of functions of more than one variable. Prerequisite: MATH 142 or the equivalent (such as: a full year of Advanced Placement Calculus or a year of junior college calculus taken as a high school student). Students entering MATH 249 should be able to differentiate and integrate functions commonly encountered in first-year calculus, including trigonometric and exponential functions, and should be able to use the derivative and the integral in common applications.
Calculus Placement Advice
Students with AP credit: A score of 4+ on the Calculus A/B exam or a 4 on the Calculus B/C exam earns credit for Math 141 and places students into Math 249 or Math 142. A score of 5 of the Calculus B/C exam earns credit for Math 141 and 142 and places students into Math 249.
Students with high school calculus but no AP credit
Calculus taken
Grades
Place into
Full year AP (A/B or B/C versions)
A's or A/B
Math 249
Full year non-AP
A's
Math 249
Full year AP (A/B or B/C versions)
B's or B/C
Math 142
Full year non-AP
A/B or B's
Math 142
Semester only or full year with lower grades
Math 141
Students wishing to place lower than recommended in this table will need departmental approval.
Students with no high school calculus who wish to enter the calculus sequence should enroll in Math 141. We recommend that they have high school math beyond Algebra II.
General Calculus Placement advice: As a rule, we recommend that students aim high in their calculus placement. If a student gets in over their head, we can help them change to a lower level course in the sequence. If a student finds him/herself unchallenged after three weeks in a lower-level course, it is often too late to change to a higher level. If in doubt, please contact the department personally.
Some majors require specific quantitative courses:
Major
Required
Biology
CHEM 116
Chemistry
CHEM 116, MATH 141* & 142*, PHYS 221 & 222
Computer Science
CS 141* & 241, MATH 142*, others
Economics
ECON 230*, MATH 141*
Exercise Science
MATH 138*
Mathematics
MATH 249* & 253
Physics
PHYS 221 & 222, MATH 249*
Psychology
PSYC 252 & 253*,
Graduate Study - GRE exam
The quantitative portion of the GRE includes some calculus-based questions
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MAT 0018 PRE – ALGEBRA Focuses on manipulative skills. Topics include prime numbers, prime factorization, the base ten number system, fractions, decimals, and integers, square roots, exponents, use of percent, formulas, measurement, properties of triangles, order of operations, basic linear equations, and introductory algebra. Designed as a preparation for MAT 0028. This course does not satisfy general education requirements in math and is awarded compensatory credit only. All students who successfully complete this course are required to complete MAT 0028 and MAT 1033 before enrolling in their first general education mathematics course and are encouraged to do so in consecutive semesters.
MAT 0020 Integrated Arithmetic and Algebra 6.00 Credits
This course combines the arithmetic and algebra skills of MAT 0012 and MAT 0024. This course includes all mathematics skills necessary for entry into college level mathematics. Arithmetic topics include operations with real numbers, fractions, decimals, exponents, geometry measurement systems, percents and ratios. Algebra topics include polynomial operation, factoring, solving and graphing linear equations and inequalities, operations with quadratic equations, and applications of all concepts. This course does not satisfy general education requirements and generates compensatory credit only.Permission from instructor required.
MAT 0028 BEGINNING ALGEBRA Provides an introduction to algebra. Topics include basic linear equations and inequalities, properties of real numbers, operations, involving exponents and polynomials, factoring, quadratic equations, applications, graphing of linear equations, and an introduction to radical simplification. This course does not satisfy general education requirements in mathematics and is awarded compensatory credit only. Credit for MAT 0024 precludes credit for MAT 0028. All students who successfully complete this course are required to complete MAT 1033 before enrolling in their first general education mathematics course and are encouraged to do so in the consecutive semester.
MAC1105 College Algebra 3 Credits
Provides students with the opportunity to gain algebraic knowledge needed for many fields such as engineering, business, science, computer technology, and mathematics. Graphical and numerical methods support the study of functions and their corresponding equations and inequalities. Students will study linear, quadratic, polynomial, rational, exponential, logarithmic, inverse, composite, radical, and absolute value functions; systems of equations and inequalities; modeling applied problems; and curve fitting techniques. Previous credit in any MAC class precludes credit in MAC 1105.
Prerequisite: MAT 1033 with a minimum grade of C.
IT IS RECOMMENDED THAT STUDENTS WHO ARE NOT REQURED TO TAKE MAC11105 FOR HTEIR PROGRAM SHOULD CONSIDER TAKING MGF1106, MGF1107, AND/OR STA2023 TO MEET THEIR 6 HOUR MATH GENERAL EDUCATION REQUIREMENT FOR AN A.A. DEGREE.
THIS IS A FAST-PACED COURSE COMBINING MAC1140 AND MAC1114 IN ONE TERM.
MAC 2233 Calculus for Business and Social Science 3 Credits
Provides a review of functions and an introduction to limits, with an emphasis on differentiation and integration of algebraic, exponential and logarithmic functions. Topics are directed toward applications in business, economic, social and behavioral sciences. Previous credit for MAC 2311 precludes credit for MAC 2233.
Prerequisite: MAC 1105 or MAC 1140 with a minimum grade of C.
MAC 2311 Calculus and Analytic Geometry I 5 Credits
This is the first of a three-course sequence in calculus. Major topics include limits, continuity, and differentiation and integration of algebraic, trigonometric, exponential and logarithmic functions. Applications include rates of change, related rates, mean value theorem, extreme values, curve sketching, differentials, area, volume and work.
Prerequisites: MAC 1114 and MAC 1140 or MAC 1147 with a minimum grade of C or appropriate score on HCC Placement Test.
MAC 2312 Calculus and Analytical Geometry II 5 Credits
This is the second in a three-course sequence in calculus. Major topics include differentiation and integration of hyperbolic functions, algebraic, trigonometric, and numerical integration techniques, applications of integrals, improper integrals, parametric equations, polar coordinates, conics, and sequences and series.Prerequisite: MAC 2311 with a minimum grade of C.
MAC 2313 Calculus and Analytical Geometry III 5 CreditsA continuation of MAC 2312. Focuses on arc length and surface area, vectors in two and three dimensional space, planes, lines and surfaces in three-dimensional space, functions of more than one variable, partial derivatives, double and triple integrals and their applications, cylindrical and spherical coordinates, vector fields, line integrals, Green's theorem and Stoke's theorem.
Prerequisites: MAC 2312 with a minimum grade of C.
MAP 2302 Differential Equations 3 Credits
Covers first order differential equations including those with separable variables, homogeneous and exact equations and equations made by an integrating factor. Topics include linear differential equations of higher order and their solutions including both homogeneous and non-homogeneous equations, differential operators, Laplace transforms, and series solutions and applications. Designed for engineering and mathematics majors.
Prerequisite: MAC 2312 with a minimum grade of C.
MGF 1106 Topics in Math 3 Credits
Topics will include finite and infinite sets, logic, deductive and inductive reasoning, geometry, counting methods, probability and statistics. Studying these topics will develop a broader base of mathematical knowledge. This course may be used to satisfy part of the mathematics general education requirement for the associate in arts degree.
Prerequisite: MAT 1033 with a C or better, or required score on HCC placement score.
MGF 1107 Explorations in math 3 Credits
Topics will be chosen from the following: financial mathematics; sequences and series; elementary number theory; history of mathematics; linear and exponential growth; voting theory; chaos and fractals; reflections and translations in geometry; graph theory; game theory; and mathematical use of calculators and computers. These topics will be helpful in developing a broader base of mathematical knowledge. This course may be used to satisfy part of the mathematics general education requirement for the associate in arts degree.
Prerequisite: MAT 1033 with a minimum grade of C or appropriate score on the HCC placement test.
STA 2023 Elementary Statistics 3 Credits
This course introduces the student to the concepts of statistical design and data analysis with emphasis on introductory descriptive and inferential statistics. Topics include data organization and analysis, probability, discrete and continuous probability distributions, confidence intervals, hypothesis testing, correlation and simple linear regression. This course may be used to satisfy part of the mathematics general education requirement for the associate in arts degree.
Prerequisites: MAT 1033 or any MAC course with a grade of C or better or appropriate score on HCC placement test.
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Saxon Math Algebra 1 4th Edition Home School Testing Book
Product description
This test book is part of Saxon's Algebra 1, 4th Edition Homeschool curriculum. Perfect for students who already have the texts, this book contains tests, a testing schedule, test answer forms, test analysis form, and test solutions. Covering every five lessons after lesson 10, tests provide opportunities to learn and practice each concept. Three optional Test Solution Answer Forms provide the appropriate workspace for students to "show their work." Answer key shows the final solution only, not the steps taken to arrive at the answer. 72 classroom-reproducible, perforated, newsprint-type pages. Softcover. 4th Homeschool Edition.
Type: Paperback ()Category: > Home SchoolingISBN / UPC: 9780547625843/0547625847Publish Date: 5/1/2011Item No: 231550Vendor: Saxon Publishers
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804-115 College Technical Math 1
Learn to solve linear, quadratic and rational equations; graphing; formula rearrangement; solving systems of equations; percent; proportions; measurement systems; computational geometry; right and oblique triangle trigonometry; trigonometric functions on the unit circle; and operations on polynomials. Emphasis will be on the application of skills to technical problems. This course is the equivalent of successful completion of College Technical Mathematics 1A and College Technical Mathematics 1B.
804-116 College Technical Math 2
Learn vectors, trigonometric functions and their graphs, identities, exponential and logarithmic functions and equations, radical equations, equations with rational exponents, dimension of a circle, velocity, sine and cosine graphs, complex numbers in polar and rectangular form, trigonometric equations, conic sections and analysis of statistical data
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books.google.com to the Theory of Weighted Polynomial Approximation
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This book serves readers who want to assess and improve the math skills they need to succeed at work. A complete review of arithmetic, algebra, geometry, and word problems ensures improvement of these essential math skills. With over 200 on-the-job practice questions, it is a necessary tool for employees who need math skills to complete their jobs with ease.
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Introduction to Actuarial and Financial Mathematical Methods
By
Stephen Garrett, Professor of Mathematical Sciences, University of Leicester, UK
This self-contained module for independent study covers the subjects most often needed by non-mathematics graduates, such as fundamental calculus, linear algebra, probability, and basic numerical methods. The easily-understandable text of Introduction to Actuarial and Mathematical Methods features examples, motivations, and lots of practice from a large number of end-of-chapter questions. Questions range from short calculations to large project-based assignments, all designed to promote independent thinking and the application of mathematical ideas. Model solutions are included.
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This Textbook
Overview
This book considers the potential and limitations of the various mathematical approaches and thereby aims to give a balanced view of the usability of each mathematical approach. Written with both student and professional in mind, this book assists the reader in applying mathematical methods to solve practical problems that are relevant to software engineers. It is suitable for coursework or self-study and there is helpful material on tools to support the various mathematical approaches.
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From the Publisher
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Handbook of Convex Geometry, Volume A offers a survey of convex geometry and its many ramifications and relations with other areas of mathematics, including convexity, geometric inequalities, and convex sets. The selection first offers information on the history of convexity, characterizations of convex sets, and mixed volumes. Topics include elementary... more...
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As a result of the editors' collaborative teaching at Harvard in the late 1960s, they produced a ground-breaking work -- The Art Of Problem Posing -- which related problem posing strategies to the already popular activity of problem solving. It took the concept of problem posing and created strategies for engaging in that activity as a central theme... more...
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The new edition of this classic book describes and provides a myriad of examples of the relationships between problem posing and problem solving, and explores the educational potential of integrating these two activities in classrooms at all levels. The Art of Problem Posing, Third Edition encourages readers to shift their thinking about problem... more...
Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other areas. The book gives an overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical source... more...
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Not a C Minus is a comprehensive study aid for senior high school Mathematics. It covers topics such as calculus, probability, finance and trigonometry, and uses a conversational, informal teaching style. Every topic is explained in detail, with sample questions and worked solutions.
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Intro
International School Almere Year planner 2008 - 2009
OVERVIEW PLANNERS – ALL SUBJECTS
MYP YEARS 2 – 5
Grade 7 - 10
September 2008
International School Almere Year planner 2008 - 2009
INDEX:
MATHEMATICS Grade 7 – 10
SCIENCE Grade 7 – 10
HUMANITIES Grade 7 – 10
LANGUAGE A
English Grade 7 – 10
Dutch Grade 7 – 10
LANGUAGE B
French Grade 7 – 10
Spanish Grade 8 – 10
Latin Grade 7 – 8
German Grade 9
TECHNOLOGY Grade 7 – 10
ART Drama Grade 7 – 10
PHYSICAL EDUCATION Grade 7 – 10
International School Almere Year planner 2008 - 2009
Subject: Mathematics Grade 7 Shapes and forms
Numbers
Locating points
Diagrams
Portfolio 2:
Lines and angles
Formulas
Measuring
Portfolio 3:
Formulas and letters
Symmetry
Simplifying and powers Grade 8 Calculating with letters
2-D Shapes
Equations
Portfolio 2:
Percentages and representing data
Square roots and pi
Pythagoras" Theorem
Portfolio 3:
General skills Quadratic equations
General skills Volumes and enlargements
Statistics and probability
Assessment:
Portfolio 1:
Unit tests
Comunication and reflection
Puzzles
Portfolio 2:
Unit tests
Comunication and reflection
The number PI
Pythagorean triples
Portfolio 3
Reflection and Evaluation
Unit tests
Clever at numbers
International School Almere Year planner 2008 - 2009
Subject: Mathematics Grade 9 Linear relationships
Similar triangles
Quadratic relationships
Portfolio 2:
Trigonometric ratios
Statistic and percentages
Inequalities and simplifying
Portfolio 3:
Trigonometry
Various relationships
Statistics and probabilities Core Numbers and patterns; Sets; Limits of accuracy
Equations- Linear equations; Construct equations from diagrams; factorizing
quadratic equations
Portfolio 2:
Angles and Similar triangles
Graphs
Portfolio 3:
Mensuration
Final revision
Assessment:
Portfolio 1:
Unit tests
Comunication and reflection
Portfolio 2:
Unit tests
Comunication and reflection
Portfolio 3
Reflection and Evaluation
Unit tests
International School Almere Year planner 2008 - 2009
Subject: Mathematics Extended Number Patterns and Ordering- Arithmetic progressions; Geometric progressions;
Tree diagram and Factorials; Permutations and Combinations
Trigonometry- Trigonometric ratios; Surds& 30, 60, 90; 45, 45, 90; Unit circle;
Trigonometric identities
Matrices- Adding and subtracting matrices; scalar and matrix multiplication; The
determinant and inverse of a matrix; Solving simultaneous equations using matrices
Algebraic relationships- Factorising quadratics; Completing the perfect square;
Simplifying fractions using factorisation
Portfolio 2:
Measurements- Perimeter; Area for squares, rectangles, parallelograms, kites,
trapeziums, circles, ovals; Heron's formula;Surface areas of common 3-D
shapes; Volume of Pyramids, prisms, cones, speheres.
Linear equations
Indices and surds
Portfolio 3:
Transforming grafs
Circle theorems; Angles; Similar triangles
Statistics- Histograms, Polygon, Circle graph; Average; Box plots; standard deviation;
Grouped data, mid-point value
Assessment:
Portfolio 1:
Unit tests
Comunication and reflection
Investigation work
Portfolio 2:
Unit tests
Comunication and reflection
Investigation work
Portfolio 3:
Reflection and Evaluation
Unit tests
Investigation work
International School Almere Year planner 2008 - 2009
Subject: Science Grade 7
MYP sciences aim to provide a worthwhile educational experience for all students whether or not they
go on to study science beyond the MYP. Our programme provides a firm foundation for further study
towards the IB Diploma.
Participation in MYP sciences should enable pupils to develop skills, understanding and knowledge of
science. They will develop an awareness of the benefits and limitations of science. Historical and
cultural contexts will enable them to see links with other subjects. Group work, particularly with
experiments, fosters international understanding and an awareness of science as a co-operative
activity. Students are encouraged to develop and apply their information technology skills.
Key Content: Assessment:
Portfolio 1: Portfolio 1:
Scientific Method Portfolio 3:
Phase Changes Evaporation lab
What is matter?
Polymers Phase change lab conclusion
Graphing activities
Tests
Portfolio 2: Use of polymers in everyday life
The Atmosphere Lab behavior
Forces
Portfolio 2:
Portfolio 3: Skin cancer project
Living vs Nonliving Things Tests
Sexual Reproduction (plant Water bottle rocket and paper
and animal) Lab behavior
Mousetrap car
Additional information
o graphing Portfolio 3:
o analyzing tables Create-a-creature model and paper
Tests
Bacteria lab
Plant growth lab
Plant reproduction cartoon
STD project
International School Almere Year planner 2008 - 2009
Subject: Science Grade 8 Classification Tests
Muscular system Posters
Skeletal system Skeletal system disorder project
Portfolio 2:
Circulatory system Portfolio 2:
Sports Injuries dissection
Digestive System sport injury poster- diagnose an
Nutrition athlete"s injury and treat-
lab behavior
Portfolio 3: tests
What is chemistry? survey of people"s diets and write a
Mixture vs compound conclusion
Portfolio 3:
Additional information Distinguishing characteristic
o graphing property labs
o analyzing tables Tests
Unknown mixture separation
Model of atoms and paper
International School Almere Year planner 2008 - 2009
Subject: Science Grade 9 Cellular Organelles Tests
Cellular Transport Osmosis lab (2x)
Cellular Respiration Role playing for cell
Photosynthesis resp/photosynthesis/ protein
Protein Synthesis synthesis
Cell respiration lab
Protein synthesis bio workbench
Portfolio 2:
What is chemistry Portfolio 2:
History of the atom Tests
Atomic structure Element research project
Bonding Ionic and covalent bonding labs
Chemical equations Acid lab
Acids and bases Acid rain research project
Chemical equation balancing
Portfolio 3: practice
Currents
Circuits Portfolio 3:
Simple machines Rube Goldberg Project
Circuit diagrams
Additional information Make-a-circuit
o graphing Tests
o analyzing tables Static electricity/current stations
International School Almere Year planner 2008 - 2009
Subject: Science BIOLOGY (compulsory) Mitosis Tests
Genetics Cancer letter
Natural selection Genetic disorder project
Evolution Evolution debate
Portfolio 2:
Portfolio 2: Tests
Ecological relationships Water quality lab
Pollution and monitoring Relationship analogies
Research on destruction of wildlife
Portfolio 3: and effect on ecological
Plant structure, relationships
Plant function, Graph analysis
Plant classification
Portfolio 3:
Flower dissection
Additional information Plant growth lab
o graphing Plant book
o analyzing tables Biome research project
o research/ citing
sources
International School Almere Year planner 2008 - 2009
Subject: Science Physics and Chemistry Percent composition tests
Moles magnesium lab
Chemical equations gum lab
Balancing equations
Chemical formulas Portfolio 2:
tests
rate of reaction lab
Portfolio 2: endothermic lab
Rates of reactions exothermic lab
Endo and exothermic factory manager role play
reactions
Portfolio 3:
Portfolio 3: tests
Radioactivity radiation presentation
Waves light and plant growth
Light connection
Sound sound wavelength vs pitch lab
Additional information
o graphing
o analyzing tables
o research/ citing
sources
International School Almere Year planner 2008 - 2009
Subject Group: Social Studies Grade 7 4 1200 BCE – 500 CE
Reading maps of time
Ancient Rome
Ancient Egypt
Ancient Greece
Geography:
o Reading maps of space
o The world through the years: land shifts and border shifts – politics, national
identity
o Land, sea, air // soil & ecosystems, coasts & shorelines, weather & climate
o The environment / problem: people and resources
End of unit test
Presentations
Poster boards
Essays and other forms of writing
Debates
Case studies
International School Almere Year planner 2008 - 2009
Subject Group: Social Studies Grade 8 settled 6 1400 – 1800 CE
Reading maps of time
Middle Ages
The Vikings
Black Death
Magna Charta
Renaissance
Reformation
Geography:
o Reading maps of space
o The world through the years: land shifts and border shifts – politics, national
identity
o Globalization
o Development
o The environment / problem: people and resources
o Global interdependence, Case studies
International School Almere Year planner 2008 - 2009
Subject Group: Social Studies Grade 9 7 1750 – 1900 CE
Maps of time
Enlightenment
Industrial Revolution
1848
(Westward) migration and emigration
Prelude to Russian Revolution
Geography:
o Reading maps of space
o The world through the years: land shifts and border shifts – politics, national
identity
o Migration / Population
o Energy
o Urbanisation
o The environment / problem: people and resources
o Human Rights,
Debates
International School Almere Year planner 2008 - 2009
Subject Group: Social Studies (compulsory) Grade 10 8 1900 – 1950 CE
o Big Era 9 1945 - Present
Reading maps of time
World War 1
Highs and Lows of the 1920"s & 1930"s
The Great Depression
Rise of Fascism / Nazism
World War 2
Cold War / McCarthyism
Vietnam
The 60"s – Gateway to a new balance of power and order
Geography:
o Reading maps of space
o The world through the years: land shifts and border shifts – politics, national
identity
o Millennium Goals
o The environment
o Human Rights
International School Almere Year planner 2008 - 2009
Subject Group: Social Studies / GBS Grade 10
General Business Studies introduces the fundamentals of Economics and modern day
business.
Key Content:
Students will study the following topics:
General Business Skills (GBS):
o Fundamentals of business
o Economic systems
o Free enterprise versus government intervention
o Public versus private sector
o Capitalism versus communism versus other forms (NL, Sweden, China, US)
o Products versus services
Entrepreneurship
o Setting up your own (service) business
o Writing a business plan
o Acquiring funding
o Where to locate?
o Human resources & the laws that cover it
o Finance / accounting
o Marketing / sales
General skills: Business plan, Dragon"s Den, End of unit test, Presentations, Poster boards
Essays and other forms of writing, Debates
International School Almere Year planner 2008 - 2009
Subject Group: Language A – English Grade 7 Telling stories
Creation Myths
Rewriting fairytales
Fantasy & Science Fiction
Patterns in 'quest' plots
Novel: The Breadwinner
Portfolio 2: POETRY
Poem as a story
Poem as a picture
Personification, Similes, Metaphors
Poem as a sound
Poem as a pattern
Novel: Skellig
Portfolio 3: NON-FICTION
Giving instructions (imperative)
Giving information
Persuading people
Autobiography
Novel: Chasing Vermeer
GRAMMAR: present, past, future, plurals, possessive, much/many
Assessment:
Portfolio 1: Portfolio 2: Portfolio 3
Present own story + Image collage Write instructions
collage Season as a person Make fact page
Write creation myth Shaping a poem Make leaflet
Write SF 'Adventure on Skellig project Write short
Venda' autobiography
Breadwinner Project Vermeer project
International School Almere Year planner 2008 - 2009
Subject Group: Language A – English Grade 8 (War) diaries & Journals
Setting scene & creating atmosphere, descriptive writing
Horror stories
Novel: Holes
Novel: How I live now
Portfolio 2: POETRY
Limericks & Riddles
Ballads
Poem as a picture, poem a s a shape
Personification, alliteration
Haiku
Rhythm, Onomatopoeia
Portfolio 3: NON-FICTION
Making news & magazines
Media images (advertising)
Graphology
Travel-writing
GRAMMAR: verb tenses, homophones, irregular verbs, mine/yours, some/any
Assessment:
Portfolio 1: Portfolio 2: Portfolio 3
Future diary Write Limerick Make newspaper on
Bill Bott's diary Image poems deadline (team)
Room description Firework poems (6 Analyze partner's
'Mystery' description different shapes) handwriting
Write horror story Holes project Write travelogue
How I live now project
International School Almere Year planner 2008 - 2009
Subject Group: Language A – English Grade 9 Stories of the world (folk tales) – patterns
Storytelling, short stories; mini sagas
Crime stories
Novel: Catcher in the Rye
Shakespeare: A Midsummer Night"s Dream
Portfolio 2: POETRY
Images, Metaphors & Feelings
Epitaphs
Dramatic monologue
Sounds, rhyme, alliteration
Villanelle
Novel: The curious incident of the dog in the nighttime
Portfolio 3: NON-FICTION
Biographies
Writing scripts
Radio commercials
TV Violence
Novel: Being
GRAMMAR: conditionals, passive voice, (in) direct speech, past perfect & continuous,
reflexive pronouns
Assessment:
Portfolio 1: Portfolio 2: Portfolio 3
Valkyries story Response to poetry Write biography
Mini saga Epitaph Write, perform script for
Letter Using sounds and radio (team)
Write Crime story alliteration in poem Debate + essay on TV
Catcher project Write villanelle violence
MSND adaptation Curious Incident project Being Project
International School Almere Year planner 2008 - 2009
Subject Group: Language A – English Grade 10 NON-FICTION
Eyewitness reports (& 'reality' shows)
Reviews
Arguments in print: Animal issues
Age of invention & scientific writing
Novel: The Book Thief
Portfolio 2: NARRATIVE
'the classics', literature time chart
Character analysis
Romantic Fiction: excerpts from Thomas Hardy (Tess), Charlotte Bronte (Jane Eyre)
Jane Austen (Pride & Prejudice) & read pre-1900 novel
Story openings
Clichés
Novel: Noughts & Crosses (0/+)
Portfolio 3: POETRY
(Extended) metaphor, personification
Writing a response/ commentary
Half and full rhyme,
Sonnets (Shakespeare, Browning, Keats)
Shakespeare: (parts of) Romeo & Juliet
Assessment:
Portfolio 1: Portfolio 2: Portfolio 3
Observation report Character Analysis Poem w/ personi- fication
Essay assignment Pre 1900 book or metaphor
Write book review presentation Write response
Animal use report Commentary Analyze poem – identify
Essay: earth in 50 yrs 0/+ project devices
Book Thief project Perform poem
Rewrite Romeo & Juliet
International School Almere Year planner 2008 - 2009
Subject Group: Language A –Dutch Grade 7The students are studying the following topics:
-Language as the most important mean to communication (oral and in writing).
-Several themes in fiction: history, the "Third World", multicultural society in works of
fiction and non-fiction.
-Language analysis (grammar), spelling, writing, reading, speaking in public and in
discussions,
listening.
-poetry
The subjects are studied in various ways. Students have to read novels and poems,
watch movies and analyse and discuss what they have seen, write stories, poems,
essays and other nonfictious texts and to give presentations (individually and in small
groups).
The used method "Op Niveau" is used as the most important mean to guide the students
along these different themes throughout the year.
Assessment:
Assessmentreuglar basis.
International School Almere Year planner 2008 - 2009
Subject: Dutch A Grade 8
Students are deepening the reading, writing and analytical skills they acquired in grade 7. A
theme of special interest in the second portfolio is "beauty". By means of "backward planning"
students are first exploring their own thoughts on this subject ("IB-student profile
characterstics "thinker" and "reflective".). After we discovered that a highly abstract concept
like beauty is best defined by describiing (some of) its characteristics, we will explore two
very influential philosophical ideas about beauty, namely that of Plato (beauty as something
we start to grasp with our ears and eyes but leads us into the realm of spiritual meaning),
Aristotle (beauty is something that set people (and, accoording to Aristotle, even things) in
motion because of its attractiveness) and John Stuart Mill (beauty-as-quality). Students are
watching movies (Finding Forrester, The Girl with a Pearl Earring) and are reflecting on this
movies using the theoretical viewpoints given to them.
During the year the students are reading at least four works of fiction and a selection of
Dutch poetry. Furthermore, they learn how to wirte a bussiness-letter and and a report. I
expect to implement a very succesful digital method called "Taalonderwijs in Ontwikkeling"
(TiO) to improve the students writing capabilties. The deepening of the student"s grammatical
knowledge and his or her active use of this knowledge in written texts, as well as the
expansion of their thesaurus thesaurus will be an important element in the grade 8
curriculum. TiO enables students to do so.
Finally, the aim is to improve the student"s reading skills to the extent that, at the end of the
year,t they are able to understand a long article of a Dutch quality newspaper (De Volkskrant
or het NRC Handelsblad)
Assessment:
Assessment takes place by several wirtten question- and answer tests. They write
book reports on the four works of fiction they have to read individually throughout the
year and are interviewed by the teacher. Presentations and group-work is also
asseseed on a regular basis. The improvement of writing- and reading skills is
measured by several individual tests.
International School Almere Year planner 2008 - 2009
Subject: Dutch A Grade 9 Grammar.
The focus is on language-analysis, spelling and thesaurus. The students learn to work with
the so called "fictie-dossier" and extend their skills in writing letters. They also learn about
different Dutch newspapers, television programms and magazines which comment on
politics, social and cultural developments.
They read a historical novel written for an adult audience.
They watch the movie ""Eline Vere".
Portfolio 2: 'Indië (the Dutch Indies)
We watch the movie "Max Havelaar" and take part in several activities in respect to the
"Multatuli"-year.
We read several fragments of Dutch authors who wrote about "Indië (the colonial name for
what is now Indonesia).
Students have to do group-presetations about a visit to the theatre or "filmhuis" (cinema for
art movies) and apply the information they got before about the use of Powerpoint.
Portfolio 3: Writing Woman, Woman on stage.
We watch several femal comedians ("caberetières) and discuss the question whether or not it
makes a difference that they are women.Students have to read a (dutch or translated) novel
from a female writer.In class we read an analyse poems from female Dutch poets (Gerhardt
and Vasalis).
International School Almere Year planner 2008 - 2009
Subject: Dutch A Grade 10 Presentation skills. Literature: Middle Ages.
Group assignment: Presentation about a sport. Students have to apply knowledge about how
to prepare and present the subject. They have to find their way through the information they
find in their book and decide for themselves when and if they need the help of the
teacher/coach.
Literature: Middle Ages: Theory about literary genres, classical reading of Karel ende
Elegast, Watching the movie Tristan and Isolde. Students read one (adapted) medieval work.
Portfolio 2: Dealing with information. Literature: 17th and early 18th century.
Students have to write an essay an show their skills in dealing assessing the value of
information. the use of sources etc.
Students watch Polanski"s adaptation of Macbeth and Molière"s L"avare with Louis de Funès
in the role of Harpagon.
They read several Dutch poems of Vondel and Hooft.
Portfolio 3. Text analysis. Literature.
The focus is on language and text analysis. Furthermore, the students learn to deal
Literature: Late 18th and 19th century.
Classical reading: Goethe"s Faust I. Music and poetry in Germany. Opera and literature.
Students have to read two literal works of this period..
International School Almere Year planner 2008 - 2009
Subject Group: Language B – French Grade 7Students introduce themselves
spell and count
give personal information (age, address, nationality, physical description)
- Grammar:
definite and indefinite articles (le,la,les / un, une, des)
the masculine and feminine of adjectives (petit/petite,gros/grosse)
the plural of nouns and adjectives (un stylo, des stylos, le livre, les livres)
presentative form (c'est/ ce sont)
the present tense of regular verbs (parler, habiter, aimer….)
Portfolio 2:
- Communication skills
talk about leisure activities
describe theirs tastes
make suggestions, accept / refuse them.
tell the time
describe daily routine (at home and at school)
- Grammar:
the present tense of irregular verbs ( être, avoir, aller, faire, pouvoir,vouloir, ….)
possessive adjectives (mon, ma, mes,ton, ta tes….)
qualificative adjectives(gender and number)
the interrogative form with …est-ce que…
the negative form (ne + verbe + pas)
prepositions (à, au, à la, au, chez, dans, en, de, …)
reflexive verbs
present continuous (etre en train de+infinitif)
Portfolio 3:
- Communication skills
describe their town
give directions
give orders
describe the weather
talk about past events
describe future plans
- Grammar:
the imperative (travaille, écoute…)
the past tense (passé composé with avoir, verbs é and some irregulars)
the future tense(proche) (aller + infinitive 8�� give personal information
describe fashion preferences
give an opinion and justify it
per - Grammar:
direct and indirect object pronouns (le, la, les)
the interrogative form (with intonation, inversion and …est-ce que…)
Questions words (pourquoi, parce que, qui, comment)
present tense (savoir,faire,prendre, aller,vouloir, pouvoir,être,avoir..)
adjectival agreement
Portfolio 2:
- Communication skills
talk and give advice about healthy eating/lifestyle
describe habits
carry out transactions (in shops, post office, station…)
talk about past and future events
- Grammar:
the future tense (aller+infinitive)
expressions of quantities and partitive articles (du, de la,de l' des)
the imperative
the past tense (passé composé with avoir) and être)
the past tense in the negative form
Portfolio 3:
- Communication skills
make predictions
locate and describe places
talk about the protection of the environment
express obligations and interdictions
- Grammar:
the past tense (passé composé with être)
the agreement of the p.p with être
futur tense (futur simple)
expressing obligations/interdicitons with il (ne) faut (pas) + infinitive and devoir
comparatives and superlatives 9 give and ask for personal details on past events
- Grammar:
present (er,ir,re)
passé composé (with être and avoir)
agreement of the past participle in passé compose with être)
the recent past (venir + infinitif)
reflexive verbs
Portfolio 2:
- Communication skills
talk about present and past habits
tell a story
talk about their educational system
talk about future events, personal projects/ professional choices
make predictions
talk about sports, hobbies and leisure activities
- Grammar:
the imparfait (regular and irregular verbs)
the opposition between passé compose and imparfait
the pronouns y and en (COI)
possessive pronouns (le/la mien/ne, le nôtre...)
Direct and indirect pronouns (le, la, les - lui, leur…)
the near future (futur proche)
the future tense
Portfolio 3:
- Communication skills
express conditions
express quantities
make suggestions
express wishes
make comparisons
- Grammar:
indefinite pronouns (ne rien , quelque chose,ne personne , quelqu'un…)
the condition (si+present, future tense)
conditional tense
comparative (plus que, moins que...)litterature, comics, songs, documentaries,
movies...)_Criteria E
International School Almere Year planner 2008 - 2009
Subject Group: Language B – French Grade 10 talk about habits and frequency
locate in time
- Grammar:
present
passé composé (with être and avoir)
agreement of the past participle in passé compose with être
dates and duration :(il y a 10 ans, en 2004,, de 1990à 1998)
imparfait
the opposition between passé compose and imparfait
Portfolio 2:
- Communication skills
talk about future events, plans
express possession
to give and justify an opinion
talk about sports, hobbies and leisure activities
- Grammar:
the pronouns y and en and relative pronouns : qui, que , où
possessive pronouns (le mien, le tien, le nôtre…)
the future tense
indefinite pronouns (ne rien , quelque chose,ne personne , quelqu'un…)
the condition (si+present, future tense)
Portfolio 3:
- Communication skills
express conditions
make suggestions, accept / refuse them
express a wish, give advice/recommendation ,make reproaches
report speech, ideas, thoughts…
- Grammar:
the conditional mood / the hypothesis ( si+imparfait, conditional)
sequence of tenses
the reported speech (past and present)
pluperfect litterature, comics, songs, documentaries,
movies...)_Criteria E
International School Almere Year planner 2008 - 2009
Subject: Language B - Spanish Grade 8Spelling and numbers
-Using the colors
-Introduce themselves
-Give personal information (age,nacionality,physical description).
Grammar
-Verb ser
-Possessive adjectives
-Feminine and masculine .Singular and plural.
-Existence
-Reflexive pronouns
Portfolio 2:
Communication skills :
-Talk about daily routine.
-Describe their houses and families.
-Talk about the weather(seasons)
-Hay
-Tell the time.
-Talk about the objects of the classroom.
Grammar
-Verb estar
-Present tense of regular and irregular common verbs
-Interrogatives
-Prepositions
-Reflexive verbs
Portfolio 3:
Communication skills
-Describe their town
-Present progressive form
-Talk about leisure activities.
Grammar
-Present progressive form :verb estar +gerundio ( ando-iendo)
-Simple past tense of regular and irregular verbs.
-Verbs: ser-estar-llevar
Assessment:
-Role playing. Criteria A-B
-Presentations. Criteria A-B
-Reading comprehension.Criteria E
-Listening comprehension.Criteria B
-Written exercises. Criteria D
-Oral questions. Criteria B
Portfolio 2:
International School Almere Year planner 2008 - 2009
Subject: Language B - Spanish Grade 9Introduce themselves
-Give information over one place.Spain
-Talk about hobbies and free time
-Talk about daily routine
Grammar
-Review from: Prepositions-reflexive pronouns and possessives
-Review of the present form of regular and irregular verbs.
-review interrogations : que –cual-como –cuantos-donde-quien
Portfolio 2:
Communication skills:
-Talk about mental and physic feelings
-Talk about like and dislike
-Present progressive
-Talk about their city, weather, clothes ,family and parts of the body
-Talk about past events
Grammar:
-Verb ser versus verb estar and tener versus estar
-Verb doler
-Simple past from regular and irregular verbs.
Portfolio 3:
Communication skills
-Talk about leisure activities
-Express frequency and quantity
-Talk about sports and animals
-Talk about places: Chile-Peru-Venezuela and mundo maya.
Grammar
Verbs gustar-encantar-preferir
Frequency and quantity adverbs.
Accentuation.
Portfolio 3
International School Almere Year planner 2008 - 2009
Subject: Language B - Spanish Grade 10icatiion skills
-Give opinion
-Agreement and disagreement
-Give instructions
-Talk about leisure activities
-Talk about their city
Grammar
-Review of : the present tense.past tense, present progressive,prepositions and
interrogatives
-Imperative form
-Verbs of opinion: parecer.creer.
-Vebos gustart,digustar,odiar.
Portfolio 2:
Communication skills
-Talk about clothes.
-Exclamation forms
-Talk about places: Colombia- Panama.
-Talk about houses and furnitures.
-Comparatives
-Talk about services and products.
Grammar
-Prepositions and adverbs of place.
-Superlatives
-Direct and indirect object
Portfolio 3:
Communication skills
-Making a recipe
-Talk about the future actions
-Talk about past actions ( imperfecto)
-Talk about obligations.
Talk about places: Guatemala, Uruguay.
Grammar
-Future tense
-Imperfecto versus preterito
-Hay que- es necesario que.
-Por-paraInternational School Almere Year planner 2008 - 2009
Subject Group: Language B – Latin Grade 7 1: Caecilius. Grammar: sentences and word order. Culture: how do Romans live.
Stage 2: In villa. Grammar: nouns, nominative and accusative case singular. Culture: Daily
life, meals
Stage 3: Negotium. Grammar: nouns and the three declensions. Culture: town of Pompeï.
Portfolio 2:
Stage 4: In foro. Grammar: the verb. Culture: the Forum.
Stage 5: In theatro. Grammar: plural, nouns and verbs. Culture: the theatre.
Stage 6: Felix. Grammar: verb imperfect and perfect tense. Culture: Slaves and freedmen.
Portfolio 3:
Stage 7: Cena. Grammar: verbs. Culture: Beliefs about life and death.
DVD: The Trojan War, the story of Odysseus.
Mythology: DVD The Storyteller
Assessment:
Portfolio 1: all stages end with a vocabulary test. The students are allowed to miss 10% of
the words. They must repeat the test until they attain this score. Only the first two tests count
for a mark. The same rules apply for the grammar/translation test.
Portfolio 2: same
Portfolio 3: same. Knowledge of the story of Homer, the Trojan war, Odysseus.
International School Almere Year planner 2008 - 2009
Subject Group: Language B – Latin Grade 8 8: Gladiatores. Grammar: plural of accusative of nouns. Culture: gladiatorial shows.
DVD (BBC): Colosseum
Stage 9: Thermae. Grammar: dative case of nouns. Culture: the baths.
Portfolio 2:
Stage 10: Rhetor. Grammar: verbs, all forms. The verb to be. Culture: Education
Stage 11: Candidati. Grammar: verbs and the dative. Questions. Culture: Local government
and elections.
Portfolio 3:
Stage 12: Vesuvius. Grammar: complete verb. Culture: the destruction and excavation of
Pompeï. DVD (BBC): Pompeï.
DVD: Jason and the Argonauts.
Students receive a certificate.
Latin in grade 9 and 10 is optional. It focuses on translations. The textbook Short Latin
Stories supplies simplified texts of main Roman writers. They are shown in the context of
history. Students also read texts in translation to familiarize themselves with the writers.
Assessment:
Assessment of vocabulary, grammar and translationskills.
International School Almere Year planner 2008 - 2009
Subject: German - Language B Grade 9
Key Content:
German is the largest language in the European Union and, being the mother tongue of very
influential thinkers and writers, the language in which many of the most influantial thoughts
and ideas that shaped the modern world are expressed.
Unfortunately, only a few students choose to follow the German course (At the beginning
of this school year only one grade 8 student showed some interest). The ISA was therefor
forced to no longer offer German as an alternative to Spanish.
The two students in grade 9 that already took German class in grade 8 are enabled to finish
this course. The same goes for the one student in grade 10 which followed German classes
in grade 8 and 9.
The curriculum for German is given by the teacher of Dutch, who lived in Germany for
several years and took several language German language courses (Deutsch als
Fremdsprache I und II). Education in German is offered in close co-operation with the
teacher of German of the grammar department of "Het Baken". To make this co-operation
easier, the decision is made to offer the German course in Dutch in order to be able to use
the same method as Het Baken does.
The aim is to get both students reacht at least at the same level as the 4 VWO students
in regular Dutch education. The one student in grade 10 has, due to his background, already
reached this level and is offered a program in German literature and "Deutschlandkunde".
Assessment:
Assessment takes place by several written question- and answer tests. They write book
reports on the four works of fiction they have to read individually throughout the year and
are interviewed by the teacher. Presentations and group-work is also assessed on a
regular basis. The improvement of writing- and reading skills is measured by several
individual tests
International School Almere Year planner 2008 - 2009
Subject: Design & Technology Grade 7 Introduction to Design Cycle
Fantasy Ice Creams
Recognition on a Medieval Battlefield
Designing travel guide
The human body
Portfolio 2: Computer Technology
Websites
Interactivity
Communication in ancient writings
Portfolio 3: 3D
Design and create a new school building
Rietveld chair
Mask
Power point presentation on Holiday destination 8 Design and exploration
My personalized Rietveld Chair
Ancient writing/expression
Design your own mosaic (Islam)
Portfolio 2: Creative art
Designing an emotional landmap
Millenium Development Goals
Designing travel guide 9 Art design
Blow-up Art
My personolized Rietveld Chair
Humanism
Portfolio 2: Food Technology
Project soft drink
Soy products affecting the world
Natural disasters
Portfolio 3: Environments
Recycling material (From cradle to cradle)
Trash Toys 10 Computer design
Create a school movie trailer
Chasing t-shirts (ppp presentation)
Portfolio 2: Food Technology
Fundraiser DRAMA Grade 7 Introduction to Drama: Listening, group & audience skills. Improvisation.
Tableau Vivante
Stage Zones & Stage Etiquette
Mime & Robot Project
Portfolio 2:
Story Telling & Devising Modern Fairy tales
Different Perspectives – The Emperor"s Clothes
Devising Drama through Teacher in Role: Darkwood Manor & King"s Fountain
Portfolio 3:
The Four Elements – Different Qualities of Movement
Ensemble Theatre
Rehearse & Perform short play: "Earth, Fire & Water"
Additional information
Particular attention is given to the keeping of a Developmental Work Book. More than half of
the assessment marks given are based on work in the DWB.
Assessment:
Portfolio 1:
Introduction to Drama : Assessment of DWB - Criteria A, C & D
Tableau: Criteria B
Stage zones: Criteria A
Portfolio 2:
Story Telling & Modern Fairy Tales: In class assessment and in DWB across all
criteria
Portfolio 3:
Four Elements & Ensemble Theatre: In class assessment and in DWB across all
criteria
International School Almere Year planner 2008 - 2009
Subject: DRAMA Grade 8 The Mantle of the Expert – Wolf Children
Creating Location – Conflict with Space
The Lost Valley – Creation of sustainable character; dance; exploring tribal worlds
Portfolio 2:
Lulu – family arguments as basis for drama. Charcterization.
Masks – working with full and half masks
Portfolio 3:
Greek Theatre
Shadow Puppets
Additional information
Particular attention is given to the keeping of a Developmental Work Book (DWB). More than
half of the assessment marks given are based on work in the DWB.
Assessment:
Portfolio 1:
Wolf Children : Criteria C
Conflict with Space: Criteria B
The Lost Valley: Assessment on written work in DWB. Criteria A, C & D
Portfolio 2:
Lulu: In class work and written work in DWB across all criteria
Portfolio 3:
Greek Theatre - Criteria A & B
Shadow Puppets: Assessment on written work in DWB. Criteria A, C & D.
Performance of puppet play: Criteria B
International School Almere Year planner 2008 - 2009
Subject: DRAMA Grade 9 Exploring social issues: Fame
Shakespeare and his world: making Shakespeare relevant to a young audience
today.
Play: Midsummer Night"s Dream
Portfolio 2:
Play: Midsummer Night"s Dream (cont)
Emotions: using and exploring emotions in drama
Advertisements
Portfolio 3:
Exploring social issues: Gender
Status: understanding and using status to create character Shakespeare – assessed across all criteria through in class assignments and written
work in DWB
Portfolio 2:
Advertisements: assessment across all criteria
Portfolio 3:
Status: assessment across all criteria
International School Almere Year planner 2008 - 2009
Subject: DRAMA Grade 10 Courtroom Drama: Understanding structure of trials as basis for drama
Frankenstein: Ethical issues in science; Frankenstein on trial, play writing skills and
stage design
Portfolio 2:
Performance: Creation of short performance pieces (one act plays or dialogues)
Portfolio 3:
Theatre of the Absurd: Becket, Pinter & Ionesco
Theatre Sport: Theatre Improvisation Frankenstein: Assessment across all criteria
Portfolio 2:
Performance: Assessment across all criteria
Portfolio 3:
Theatre of the Absurd: Criteria A & B
Theatre Sport: Criteria B, C & D
International School Almere Year planner 2008 - 2009
Subject group: Physical Education Grade 7 (introduction) Dancing Around the world Group: Physical Education Grade 8 High Jump Creative Dance 9 sporting Acrobatics, transfer of weight
Athletics Mini-competition
Running
Portfolio 2:
Hockey
Volleyball
Running
Condition Circuit
Portfolio 3:
Basketball
Racketsports
Trapeze
Aerobics 10lop
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About this item
Volume II of a two-part series, this book features 74 problems from various branches of mathematics. Ranging from relatively simple to extremely difficult, their topics include points and lines, topology, convex polygons, theory of primes, nondecimal counting, and other subjects. Suitable for students, teachers, and any lover of mathematics. Complete solutions
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MAZ5025: Financial Literacy (2013-2014)
Major Concepts/Content: This course introduces students to the mathematics and mathematical models used in various financial topics. The focus will be on the applied mathematics, primarily algebraic concepts, surrounding finance and business fundamentals. Students are provided opportunities to develop habits of mind while applying skills and knowledge in mathematics to the area of Finance. Concepts related to graphical, tabular, and algorithmic representations of functions introduced in Algebra I will be reinforced and enhanced through the financial problems explored.
Major Instructional Activities: Students will be expected to complete a number of research-based projects, differentiated to provide relevance to their learning. Working together, students will problem solve making connections to the world around them as well as to other math concepts. Students will practice effective oral and written communication as they explain and justify their thinking. Students will be given opportunities to explore technology and multimedia to stimulate their imagination and curiosity about applied mathematics in the area of Finance. The classroom is expanded to include the greater community as self-directed students work both independently and interdependently.
Major Evaluative Techniques: Performances, projects and multiple forms of media are used for self, peer and other forms of assessment. Grades are based on what a student has learned.
Course Objectives: Upon completion of this course, students will:
Understand and use exponential growth models to determine present and future value of a single deposit.
Quantify and understand the opportunity costs of making financial decisions.
Use multiple representations of mathematical models, primarily algebraic, to make informed decisions.
Compare and contrast the costs of checking account options using algebraic models.
Create and interpret mathematical models that represent the cost of borrowing money on various loan products.
Use systems of equations to compare and contrast varying loan options on mortgages and other financial products.
Understand and explain the applied relevance of the mathematical models created in context of a problem being solved.
Understand the mathematics behind annual percentage rate (APR) and annual percentage yield (APY) calculations and the related change as a result of various bank fees.
Understand the proportional relationship between interest rate and risk on varying types of investments and loan products
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Everyday Math Demystified [NOOK Book],...
More About
This Book, and much more.
Related Subjects
Meet the Author
Stan Gibilisco is one of McGraw-Hill's most prolific and popular authors. His clear, reader-friendly writing style makes his electronics books accessible to a wide audience, and his background in mathematics and research makes him an ideal editor for professional handbooks. He is the author of the TAB Encyclopedia of Electronics for Technicians and Hobbyists, Teach Yourself Electricity and Electronics, and The Illustrated Dictionary of Electronics. Booklist named his McGraw-Hill Encyclopedia of Personal Computing a "Best Reference" of 1996ensChan
Posted June 26, 2012
Good reference to many squares in the place of math symbols in examples...
Great book!!!!
Though I find that the publisher did not take the time to
create a lot of the mathematical symbols in formulas; there are a lot of squares (example: Aset [] Bset)--this is the symbol used to indicate the symbol is unrecognized by, in this case an ipad 1st generation--that requires the reader to go back to the math symbols chart many, many times to figure out what symbol should be visible. Out side of those squares in math formulas this is a great reference.
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Anonymous
Posted June 6, 2004
What Every Adult Should Know
As an educator (retired), I got an advance copy of this book. My first reaction was, 'This is everyday math?!' Then in the third chapter it hit me: This is not necessarily what everyone knows (if that was the case, there would be no need for the book). It's what every American adult should know by the time they graduate from high school. Sadly, given the state of math education in this country, this book probably should have the subtitle 'in an Ideal World.' I recommend that anyone who wants to really understand math, and not just rush through it as some sort of evil necessity, study this book thoroughly -- after, or in addition to, their high school courses. And don't fret the abstract stuff. Math is abstract by its very nature.
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| 677.169 | 1 |
At the beginning of the twentieth century, college algebra was taught differently than it is nowadays. There are many topics that are now part of calculus or analysis classes. Other topics are covered only in abstract form in a modern algebra class on field theory. Fine's College Algebra offers the reader a chance to learn the origins of a variety of topics taught in today's curriculum, while also learning valuable techniques that, in some cases, are almost forgotten.
In the early 1900s, methods were often emphasized, rather than abstract principles. In this book, Fine includes detailed discussions of techniques of solving quadratic and cubic equations, as well as some discussion of fourth-order equations. There are also detailed treatments of partial fractions, the method of undetermined coefficients, and synthetic division.
The book is ostensibly an algebra book; however, it covers many topics that are found throughout today's curriculum:
Though the book is structured as a textbook, modern mathematicians will find it a delight to dip into. There are many gems that have been overlooked by today's emphasis on abstraction and generality. By revisiting familiar topics, such as continued fractions or solutions of polynomial equations, modern readers will enrich their knowledge of fundamental areas of mathematics, while gaining concrete methods for working with their modern incarnations. The book is suitable for undergraduates, graduate students, and researchers interested in algebra.
Readership
Undergraduates, graduate students, and research mathematicians interested in algebra.
Reviews
"The author has arranged a great variety of classical, elementary material in a very original manner, which every college student or grammar-school master can still considerably profit from, even so in these days."
-- Zentralblatt MATH
From a review of the previous edition:
"This book contains more than would seem possible from the title ... the author demonstrates that he is taking pains to bring scientific rigor into accord with pedagogical considerations."
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Alumni of our math program have been very successful. Our alumni profiles feature some of their positions, including Actuary, Medical Doctor, Lawyer, High School Teacher and Vice President of Information Management.
The five professors in the our math department have a wide variety of mathematical interests, including game theory, mathematical modeling, statistics, chaos theory, geometry, knot theory, graph theory, and differential equations. They are also interested in interdisciplinary applications of math in fields such as political science, economics, education, biology, chemistry and physics.
Why Simpson?
The Great Value Colleges website ranked Simpson College No. 7 in its list of "50 Great Affordable Private Colleges in the Midwest," the highest ranking of any public or private college or university in Iowa.
| 677.169 | 1 |
Prepare for success in mathematics with DOING MATHEMATICS: AN INTRODUCTION TO PROOFS AND PROBLEM SOLVING! By discussing proof techniques, problem solving methods, and the understanding of mathematical ideas, this mathematics text gives you a solid foundation from which to build while providing you with the tools you need to succeed. Numerous examples, problem solving methods, and explanations make exam preparation easy.
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Most Helpful Customer Reviews
This is a swell introduction to analysis. Plenty of good examples, with a well-ordered approach to the material. I particularly appreciated the section on the mindsets and assumptions that affect how we do mathematics. It's really helpful in seeing that mathematics is creative and intuitive, and not as bluntly algorithmic as so many high school classes seem. A quality read.
This is an outstanding book that teaches mathematics from the ground up, starting with elementary logic and working its way up gradually through the techniques and notation needed to formulate and rigorously prove theorems. Along the way, it touches on the fundamentals of set theory, number theory, calculus, analysis and linear algebra. There are no prerequisites, making the book suitable for everyone from precocious high schoolers to graduate students.
I had the pleasure of using the earlier edition of this book ("Introduction to Mathematical Structures," 1989) for the logic and proofs course required of math majors at Carleton College. After this edition came out, I became the grader for the course. I found the problems in the book to be very well-crafted, challenging yet approachable for the students. Nearly all of them felt both necessary and sufficient.
The book is much-improved in its second edition, with a generous number of examples and exercises. I would recommend it to anyone pursuing or even contemplating a degree in mathematics, science or philosophy.
I am astonished by the other reviews. Obviously these reviewers are far more proficient in mathematics than I. However, I feel that I am on strong ground when I disagree with the notion that this book would make a good introductory text for undergraduates or high school students. Many parts of the text are not clear. Often the author's explanation of defnitions (which are usually easy to understand) create confusion. Further, I do not think starting out with truth tables is particularly appropriate. Better that the author set the context for valid and invalid arguments, then implications, then the details of the truth tables. Chapter 4 on Set Theory is generally horrible, particularly the subject of Relations. (The reader would be better served by relying on the classics such as Halmos.) I found that Wikipedia is much more illuminating and clearer than Galovich. In general, using Galovich drove me to rely on Wikipedia, Mathworld, etc more than any other math text.
In conclusion, I think the author tried to cover too many subjects in 300 pages. Also I think the layout and organization of the book needs some work.
| 677.169 | 1 |
More About
This Textbook
Overview
An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems. Based on their teaching experiences, the authors offer an accessible text that emphasizes the fundamentals of discrete mathematics and its advanced topics. This text shows how to express precise ideas in clear mathematical language. Students discover the importance of discrete mathematics in describing computer science structures and problem solving. They also learn how mastering discrete mathematics will help them develop important reasoning skills that will continue to be useful throughout their careers.
Related Subjects
Meet the Author
Gary Haggard is Professor of Computer Science at Bucknell University. His research in data structures focuses on the implementation of effective algorithms for computing invariants for large combinatorial structures such as graphs. Dr. Haggard's current work is directed towards finding chromatic polynomials of large graphs.
John Schlipf is a Professor of Computer Science in the Department of Electrical and Computer Engineering and Computer Science at the University of Cincinnati. His research interests include logic programming and deductive databases, algorithms for satisfiability, computability and complexity, formal verification, and model theory.
Sue Whitesides is Professor of Computer Science at McGill University. She holds a Ph.D. from University of Wisconsin and a Masters from Stanford University. Her research interests lie within combinatorial mathematics and theoretical computer
| 677.169 | 1 |
School teachers need a way to efficiently evaluate their students, and this can be done by grading them based on their answers to certain assignments and exams.
Tailored for math teachers, Infinite Algebra 1 is a handy piece of software that can help you generate tests that contain various mathematical questions and equations. The application requires a stable Internet connection in order to properly work
Comprehensive math quiz generator
The program allows you to create worksheets containing various math quizzes covering a wide range of equations and mathematical theorems. This way, you do not have to manually compose and write it in electronic format, as this task takes a lot of time, especially if you need to make a different test for each student. For this, you would need a wide range of exercises and questions.
Because the application can generate other versions if the same question, you can create customized tests for each student in particular.
Intuitive math assignment creator
Infinite Algebra 1 provides you with an efficient and convenient way of creating math exams for any of your students, regardless of their level of math training and understanding. Each created worksheet can be rearranged on the spot, allowing you to generate professional-looking exams without effort.
Furthermore, you can export any of the created worksheets, along with its answer sheet to PDF, which is an ideal format for document printing. Doing so enables you to easily correct the exams of your students, and properly grade them.
An overall good exam generator
To conclude, Infinite Algebra 1 helps people who want to hone their mathematical understanding and teachers alike to create quiz tests that cover most algebraic theorems and equations. These questions have various difficulty, meaning that they can be used to test both advanced or beginner students with ease.
Infinite Algebra 1 was reviewed by Andrei Fercalo, last updated on March 24th, 2015
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2009, with emphasis on engineering design.
Extensive Use of computational tools: Matlab sections at end of each chapter show how to implement concepts from the chapter.
Frees the student from the drudgery of mundane calculations and allows him to consider more subtle aspects of control system analysis and design.
An engineering approach to digital controls: emphasis throughout the book is on design of control systems. Mathematics is used to help explain concepts, but throughout the text discussion is tied to design and implementation. For example coverage of analog controls in chapter 5 is not simply a review, but is used to show how analog control systems map to digital control systems.
Review of Background Material: contains review material to aid understanding of digital control analysis and design. Examples include discussion of discrete-time systems in time domain and frequency domain (reviewed from linear systems course) and root locus design in s-domain and z-domain (reviewed from feedback control course).
Inclusion of Advanced Topics In addition to the basic topics required for a one semester senior/graduate class, the text includes some advanced material to make it suitable for an introductory graduate level class or for two quarters at the senior/graduate level. Examples of optional topics are state-space methods, which may receive brief coverage in a one semester course, and nonlinear discrete-time systems.
Minimal Mathematics Prerequisites The mathematics background required for understanding most of the book is based on what can be reasonably expected from the average electrical, chemical or mechanical engineering senior. This background includes three semesters of calculus, differential equations and basic linear algebra. Some texts on digital control require more mathematical maturity and are therefore beyond the reach of the typical senior
| 677.169 | 1 |
Tough Test Questions? Missed Lectures? Not Enough Time?
Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in... more...
Review of Elementary Mathematics , Second Edition, assumes no prior knowledge of mathematics and makes accessible all of elementary math, including the basics of algebra and geometry. It features up-to-date terminology plus added explanations of how to make calculations both with and without a calculator. Thousands of questions, with fully worked... more...
Perfect for high-school seniors and college freshman
Covers the fundamentals of basic college mathematics
Reflects the newest curriculum
Over 1500 solved problems
Use with these courses:
College Algebra
Trigonometry
Discrete Mathematics
Pre-Calculus
Calculus
Introduction to Mathematic Modeling
SCHAUM'S OUTLINES... more...
Phillip A. Schmidt, Ph.D. , is the program coordinator for secondary education at the Teachers College of Western Governors University.
Robert Steiner, Ph.D. , is the project director of seminars on science at the American Museum of Natural History. more...
The late Frank Ayres, Jr., Ph.D. , was formerly a professor in and head of the Department of Mathematics at Dickinson College, Carlisle, Pennsylvania. He is the author or coauthor of eight Schaum's Outlines, including Calculus, Trigonometry, Differential Equations, and Modern Abstract Algebra.
Philip A. Schmidt, Ph.D. , has a B.S. from Brooklyn... more...
Philip A. Schmidt, Ph.D. is the director of the Mathaware Project, at Berea College. He is the author of several Schaum's Outlines.
Dr. Barnett Rich held a Ph.D. from Columbia University and a J.D. from NYU. Dr. Rich was the author of 23 books. more...
Tensors and methods of differential geometry are very useful mathematical tools in many fields of modern physics and computational engineering including relativity physics, electrodynamics, computational fluid dynamics (CFD), continuum mechanics, aero and vibroacoustics, and cybernetics.This book comprehensively presents topics, such as bra-ket notation,... more...
The book's objective is to provide an easy-to-read, systematically arranged scientifically practical textbook. In addition to students, it also serves judges, lawyers and civil servants as a valuable aid, which offers accuracy and clarity despite the abundance of subject material. The law card catalogue is added as CD-ROM in the current 2005 version:... more...
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Algebraic Lit Pathway
Math 98 (U; 0 SCH); Math 108 (U; 3 SCH)
"I love that I can watch the video lectures many times and understand things on my own terms."
The Algebraic Literacy Pathway prepares your for mathematics futures that include business calculus (Math 211), pre-calculus (Math 116/117) leading to calculus (Math 231), 100-level Chemistry courses, and other courses requiring a background beyond the Mathematical Literacy Pathway. The focus of this Pathway is on building understanding of mathematical systems, but emphasizing not only the symbolism and manipulation that you might remember from high school, but also modeling and application that will build the conceptual knowledge that you need to succeed. The Algebraic Literacy Pathway includes quantitative topics from areas besides algebra, which supports the needs of both STEM (Science, Technology, Engineering, and Math) bound students and other students, and satisfies the General Education/Quantitative Literacy Part A requirement for graduation.
The sample lesson to the left shows how this all works – real world examples that demonstrate why concepts are important, but enough theory to build the base you need for future success.
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More About
This Book
Overview
Considered the standard quick-reference tool for students of mathematics, The Penguin Dictionary of Mathematics has been completely revised, updated, and expanded. It now includes:
* over 3,000 headwords, fully cross-referenced and with approp-riate diagrams
* entries that cover all branches of pure and applied mathematics
* biographies of over 200 key figures in mathematics
* over 350 new entries on the latest topics such as chaos, fractals, and graph theory 16, 2003
...is this out of print?
this book explains many concepts in a very digestable way.., I'm hoping it hasn't gone the way of so many other Mathematics Dictionaries in the last few years, good books with good illustration - out of print!
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Anonymous
Posted April 20, 2001
Nifty
It's great if you are delving into material at a rapid pace and thus don't have the time to completely remember certain notions untill you formally learn them. I take this thing everywhere with me! It is great and small and perfect for an acceling beginning undergrad or even a pretty advanced high schooler. It does lack a lot of advanced definitions though, such as anything about most of the clay mathematics millenium problems
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MATH001 PZ-01 — Math Philosophy & the Real World
Grabiner, Judith — credit: 1.0
departments: Science,Technlgy,Society, Mathematics
Letter Grade
Throughout history, mathematics has changed the way people look at the world. This course will focus on two examples: Euclidean geometry (which suggested to philosophers that certainty was achievable by human thought) and probability and statistics (which gave scientists a way of dealing with events that did not seem to follow any laws but those of chance). Readings and problems will be taken from three types of sources: (1) Euclid's Elements of Geometry; (2) modern elementary works on probability and its applications to the study of society and to gambling; (3) the writings of philosophers whose views were strongly influenced by mathematics, such as Plato, Aristotle, Pascal, Spinoza, Kant, Laplace, Helmholtz, and Thomas Jefferson. Prerequisite: high school algebra and geometry.
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An introduction to the basic techniques and functions of mathematics. This course is especially recommended for those students who need to brush up due to a shaky high school preparation or for those who haven't had a mathematics course in several years. Topics include linear equations and inequalities, quadratic equations, polynomials, and rational functions and their inverses, including the exponential and the logarithm. Note: Not open to students who have passed MATH 150 or above. Not transferable to other Maryland public institutions for college-level credit. Prerequisite: Qualifying score on LRC Algebra placement exam. For more information contact sriley4@umbc.edu.
An introduction to linear algebra, matrices, set theory, combinatorial analysis and probability theory. Appropriate for students desiring knowledge of elementary linear algebra and probability theory. Note: Not open to students who have passed MATH 221, STAT 350, STAT 351, STAT 355 or STAT 451. Prerequisite: A qualifying score on the LRC Algebra placement test. For more information contact ioana1@math.umbc.edu.
This course provides the mathematical preparation necessary for success in calculus. It also provides preparation for basic physics, computer science and engineering science courses. Topics covered include review of functions and graphing techniques, logarithmic and exponential functions; review of basic right-angle trigonometry followed by an extensive treatment of trigonometric functions, identities and applications to the analytic geometry of the conic sections, applications to two-dimensional vectors and to the geometry of complex numbers. Prerequisites: A suitable score on LRC Algebra placement exam or MATH 106. For more information contact dwang2@math.umbc.edu.
Topics of this course include: limits, continuity, the rate of change, derivatives, differentiations formulas for algebraic and trigonometric functions, maxima and minima, integration and computation of areas. Areas and volumes of solids of revolution, applications. Note: Non-science-oriented students interested in calculus should consider MATH 155. Credit will not be given for both MATH 151 and 155. Prerequisite: MATH 150 or a qualifying score on the LRC Calculus Readiness placement test. For more information for (6080) contact olenshe1@math.umbc.edu; (6082) mcwil@comcast.net.
Topics include: logarithmic and exponential functions, inverse functions, methods of integration, improper integrals, hyperbolic functions, sequences and infinite series, power series, Taylor series, applications, conic sections and polar coordinates. Note: There is also a separate discussion time, which is recommended but not required. A teaching assistant will lead the discussion to assist students in understanding the material. Prerequisite: MATH 141, MATH 151 or MATH 155B. For more information contact (6080): tighe@math.umbc.edu.; (6081): kallen1@math.umbc.edu.
Basic ideas of differential and integral calculus, with emphasis on elementary techniques of differentiation and integration with applications are treated in this course. Not recommended for students majoring in mathematics, computer science, engineering, biological or physical sciences. Note: Credit will not be given for both MATH 151 and 155. Prerequisite: A suitable score on the LRC Algebra placement test or MATH 106. For more information contact wfs-iii@comcast.net.
Introduction to convex sets. Theory of linear programming. Applications to transportation and assignment problems. Introduction to graphs with applications to network problems, including shortest route and maximum flow problems. Introduction to game theory. Note: Credit will not be given for both MATH 380 and MATH 381. Prerequisite: MATH 221.
The student will become familiar with the usage of Matlab, an advanced numerical linear algebra package that is widely used in teaching and research. Matlab is an interactive tool for high- performance numerical computations, visualization and programming. Matlab performs complex matrix algebra, computes matrix factorizations (such as LU, QR and SVD) and eigenvalues, solves linear systems of equations, provides extensive 2D and 3D visualization tools, and possesses programming tools used in scripts and functions. Prerequisites: MATH 152, 221 and CMSC 201, or permission of instructor. For more information see
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Web Site Order of Operations Mrs. Glosser's Math Goodies (Unit 7) Students can interpret a problem differently, resulting in two different answers. Mathematicians have devised a standard... Curriculum: Mathematics Grades: 4, 5, 6, 734.
Web Site S.O.S. Mathematics - Calculus Check out a good list of calculus problems with solutions. This is a free resource for math review material from Algebra to Differential Equations!
Web Site Order of Operations When a numerical expression involves two or more operations, there is a specific order in which these operations must be performed. The phrase PEMDA (Parenth... Curriculum: Mathematics Grades: 5, 6, 7, 8
39.
Web Site Intermediate Algebra Tutorials 42 Tutorials that math teachers can use with student or students can work on their own to reinforce skills, as homework, or review during class. Tutorials in... Curriculum: Mathematics Grades: 6, 7, 8, 9, 10, 11, 12, Junior/Community College, University
40.
Web Site Variables This site covers symbol variables and substitution of symbols to discover unknown values. In simple terms it shows you how a box is waiting for a value. (Key... Curriculum: Mathematics Grades: 6, 7, 8
By Resource Type:
Web Site Document or Handout Image Template Book Video
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Trigonometry
Browse related Subjects
...Read More illustrations drawn from Lance Armstrong's cycling success, the Ferris wheel, and even the human cannonball show trigonometry in action. Unique Historical Vignettes offer a fascinating glimpse at how many of the central ideas in trigonometry began. TRIGONOMETRY 6e, International Edition, uses a standard right-angle approach with an emphasis on the study skills most important for success both now and in advanced courses, such as calculus. The book's proven blend of exercises, fresh applications, and projects is combined with a simplified approach to graphing and the convenience of new Enhanced WebAssign--a leading, time-saving online homework tool--and the innovative CengageNOW teaching system. With TRIGONOMETRY 6e, International Edition, you'll find everything you need for a thorough understand of trigonometry concepts now and the solid foundation you need for future coursework and career success.Read Less
Good. 1285032144. This book is in good condition only. The book has some shelfwear, edgewear, corner wear, light cover creasing. Inside pages are clean. Solid study or reading copy but not for collectors.; 10 X 7.90 X 0.90 inches; 592 pages.
Fair. Hardcover. All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing. SKU: 9781111826857-5Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9781111826857-4Very good. Hardcover. Has minor wear and/or markings. SKU: 9781111826857-3Fair. 1111826854
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24Elementary Differential Equations with Boundary Value Problemsintegrates the underlying theory, the solution procedures, and the numerical/computational aspects of differential equations in a seamless way that provides students with the necessary framework to understand and solve differential equations. Theory is presented as simply as possible with an emphasis on how to use it. With an emphasis on linear equations, linear and nonlinear equations (first order and higher order) are treated in separate chapters. In developing mathematical models, this text guides the student carefully through the underlying physical principles leading to the relevant mathematics. Asking students to use common sense, intuition, and 'back-of-the-envelope' checks as well as challenging them to anticipate and interpret the physical content of the solution encourage critical thinking. MARKET: Intended for use in introductory course in differential equations that includes boundary value problems.
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Study Notebook, CCSS
Quick Review Math Handbook, Book 3, Spanish Student Edition
Algebra 1, Noteables: Interactive Study Notebook with Foldables
Algebra 1, StudentWorks Plus DVD-ROM
Algebra 1, Spanish Study Guide and Intervention Workbook
Algebra 1, Spanish Practice Workbook
Algebra 1, Spanish Word Problems Practice Workbook
Algebra 1, Study Guide and Intervention Workbook
Algebra 1, Practice Workbook
Algebra 1, Skills Practice Workbook
Summary
TheStudy Notebookcontains a note-taking guide for every lesson in the Student Edition. This notebook helps students: Preview the lesson, Build their mathematics vocabulary knowledge, Organize and take notes using graphic organizers, Increase their writing skills, and Prepare for chapter tests.
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More About
This Textbook
Overview
Until recently, almost all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming.
Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs. In this book you will find an introduction to GA that will give you a strong grasp of its relationship to linear algebra and its significance for your work. You will learn how to use GA to represent objects and perform geometric operations on them. And you will begin mastering proven techniques for making GA an integral part of your applications in a way that simplifies your code without slowing it down.
* The first book on Geometric Algebra for programmers in computer graphics and entertainment computing
* Written by leaders in the field providing essential information on this new technique for 3D graphics
* This full colour book includes a website with GAViewer, a program to experiment with GA
Related Subjects
Meet the Author
Daniel Fontijne holds a Master's degree in artificial Intelligence and a Ph.D. in Computer Science, both from the University of Amsterdam. His main professional interests are computer graphics, motion capture, and computer
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A non-traditional Algebra text (high school and early college levels) placed on the Web by the Science Education Team at Los Alamos National Laboratory. Browse it on the Web or download a PDF version. Chapter headings...
Produced by Science Academy Software, this site is a collection of math questions on subjects including basic arithmetic, order of operations, calculating perimeters and distance, exponents, and bar graphs. It is an...
The introduction to this site remarks, "If you need help in college algebra, you have come to the right place." Their statement is accurate, as the staff members at the West Texas A&M University's Virtual Math Lab have...
Presented by HippoCampus, a project of the Monterey Institute for Technology and Education, this free online course "is a study of the basic skills and concepts of elementary algebra, including language and operations...
Presented by Professor Jody Harris at Broward Community College and created by Professor Levy, these handouts are an excellent resource to print and give to community and technical college students in the algebra...
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Elementary and Intermediate Algebra for College Students
Browse related Subjects
...Read More procedures. Understanding Algebra call-outs highlight key points throughout the text, allowing readers to identify important points at a glance. The updated examples use color to highlight the variables and important notation to clearly illustrate the solution process Hardcover. Has minor wear and/or markings. SKU: 9780130139801-3-0-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions. ISBN: 9780130139801.
Fair. Hardcover. All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing. SKU: 9780132337229-5-0-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions. ISBN: 9780132337229great value
The book was in good condition and better yet the price was super good. Also received the book in the proper timing.
mathbook
Oct 11, 2007
Math Book
This book is a good book to start off with if you don't really understand math that well. the only problem with the book i reviced was the teacher edition and i asked for the student. but everything else is good
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Course Details
Select Term:
Mathematics I for Technicians and Technologists MATH1131 P
Units:
4
Description:
In this course you will learn the mathematics required for your first year program courses. You will focus on many electrical applications including DC circuits, RL and RC charging circuits, series AC circuits and digital circuits. You will also acquire valuable problem-solving tools and the skills necessary for the remaining mathematics courses in the technology series.
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Quoted from the site: [This site contains...] "Free mathematics tutorials to help you explore and gain deep understanding of...
see more
Quoted from the site: [This site contains...] "Free mathematics tutorials to help you explore and gain deep understanding of math topics." The math topics covered include 1) Precalculus Tutorials 2) Calculus Tutorials and Problems 3) Geometry Tutorials and Problems 3) Trigonometry Tutorials and Problems for Self Tests 4) Elementary statistics and probability tutorials 5) Applications of mathematics in physics and engineering. And much more, including many Tutorials and Problems (with applets) to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Mathematics Tutorials and Problems (with applets)
Select this link to open drop down to add material Mathematics Tutorials and Problems (with applets) to your Bookmark Collection or Course ePortfolio
This subsite of Mathematics Tutorials and Problems (with applets) is divided into Interactive Tutorials, Calculus Problems,...
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This subsite of Mathematics Tutorials and Problems (with applets) is divided into Interactive Tutorials, Calculus Problems, and Calculus Questions, Answers and Solutions. Here the user will find applets with guided exercises and many examples and worked out problems applicable to the first year of Calculus Tutorials and Problems to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Calculus Tutorials and Problems
Select this link to open drop down to add material Calculus Tutorials and Problems to your Bookmark Collection or Course ePortfolio
This site is devoted to learning mathematics through practice. Many dozens of practice problems are provided in Precalculus,...
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This site is devoted to learning mathematics through practice. Many dozens of practice problems are provided in Precalculus, Calculus I - III, Linear Algebra, Number Theory, and Abstract Algebra. The last two subject areas -- referred to as "books" on the site -- are under construction. To each topic within a book (for example, Epsilon and Delta within Calculus I) there is a "module" of approximately 20 to 30 problems. Each module also includes a help page of background material. The modules are interactive to some extent and often provide suggestions when wrong answers are enteredOW -- Calculus on the Web to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material COW -- Calculus on the Web
Select this link to open drop down to add material COW -- Calculus on the Web to your Bookmark Collection or Course ePortfolio
This site consists of examples, exercises, games, and other learning activities associated with the textbook, Discrete...
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This site consists of examples, exercises, games, and other learning activities associated with the textbook, Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns and Games by Doug Ensley and Winston Crawley. Requires Adobe Flash player Math Resources to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Discrete Math Resources
Select this link to open drop down to add material Discrete Math Resources to your Bookmark Collection or Course ePortfolio
This site consists of several dozen applets, each pertaining to a different aspect of single variable Calculus. Topics...
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This site consists of several dozen applets, each pertaining to a different aspect of single variable Calculus. Topics covered range from continuity and limits to differential equations and infinite series. Each applet is on a separate page that also includes detailed explanations of the concept under review along with suggestions for student experimentation with the applet. Each applet can be opened in a resizable window for better viewing as a classroom demonstration Calculus Applets to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Introduction to Calculus Applets
Select this link to open drop down to add material Introduction to Calculus AppletsQuoted from the site: "Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in...
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Quoted from the site: "Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, and more Real Analysis to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Interactive Real Analysis
Select this link to open drop down to add material Interactive Real Analysis
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...I have also taken graduate-level math coursework at the Johns Hopkins University Applied Physics Lab. I have worked for over 40 years as an engineer applying math to the solution of real-world problems. Algebra is the abstraction of basic arithmetic, using letters to stand in for specific known or unknown numbers
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Just for you, we curate a growing list of independent booksellers, giving you even MORE choices when shopping for your textbooks.Keep in mind: Marketplace orders do NOT qualify for free shipping. More about the Marketplace
Brand new. We distribute directly for the publisher. The fundamental idea of geometry is that of symmetry. With that principle as the starting point, Barker and Howe begin an insightful and rewarding...show more study of Euclidean geometry.The primary focus of the book is on transformations of the plane. The transformational point of view provides both a path for deeper understanding of traditional synthetic geometry and tools for providing proofs that spring from a consistent point of view. As a result, proofs become more comprehensible, as techniques can be used and reused in similar settings.The approach to the material is very concrete, with complete explanations of all the important ideas, including foundational background. The discussions of the nine-point circle and wallpaper groups are particular examples of how the strength of the transformational point of view and the care of the authors' exposition combine to give a remarkable presentation of topics in geometry.This text is for a one-semester undergraduate course on geometry. It is richly illustrated and contains hundreds of exercises. ...show less
2007 Hardback NEAR FINE 9780821839003
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Focused on helping students understand and construct proofs and expanding their mathematical maturity, this best-selling text is an accessible introduction to discrete mathematics. Johnsonbaugh's algorithmic approach emphasizes problem-solving techniques. The Seventh Edition reflects user and reviewer feedback on both content and organization.
CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
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This books gives an introduction to discrete mathematics for beginning undergraduates. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. Many examples of formal and informal proofs are given. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. The rest of the book is more standard. It deals with functions and relations, directed and undirected graphs, and an introduction to combinatorics. There is a section on public key cryptography and RSA, with complete proofs of Fermat's little theorem and the correctness of the RSA scheme, as well as explicit algorithms to perform modular arithmetic. The last chapter provides more graph theory. Eulerian and Hamiltonian cycles are discussed. Then, we study flows and tensions and state and prove the max flow min-cut theorem. We also discuss matchings, covering, bipartite graphsThis well-written, highly illustrated book will be very useful and interesting to students in both mathematics and computer science. … Attractive features of this book include clear presentations, end-of-chapter summaries and references, a useful set of problems of varying difficulty, and a symbol as well as a subject index. Summing Up: Highly recommended. Upper-division undergraduates, graduate students, and professionals/practitioners." (D. V. Chopra, Choice, Vol. 48 (11), July, 2011)
"This book is intended to be a textbook for students in Computer Science, covering basic areas of Discrete Mathematics. … lots of references to supplementary or more advanced literature are provided, and less basic and more sophisticated problems as well as connections to other areas of science are given. Each chapter closes with a rich collection of exercises, which often include hints to their solution and further explanations." (Martina Kubitzke, Zentralblatt MATH, Vol. 1227, 2012)
"This book provides a rigorous introduction to standard topics in the field: logical reasoning, sets, functions, graphs and counting techniques. Its intended audience is computer science undergraduate students, but could also be used in a course for mathematics majors. … Each chapter has a summary and a generous number of exercises … . The exposition is structured as a series of propositions and theorems that are proved clearly and in detail. Historical remarks and an abundance of photographs of mathematicians enliven the text." (Gabriella Pinter, The Mathematical Association of America, February, 2012)
From the Back Cover
This book gives an introduction to discrete mathematics for beginning undergraduates and starts with a chapter on the rules of mathematical reasoning.
This book begins with a presentation of the rules of logic as used in mathematics where many examples of formal and informal proofs are given. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. The rest of the book deals with functions and relations, directed and undirected graphs and an introduction to combinatorics, partial orders and complete induction. There is a section on public key cryptography and RSA, with complete proofs of Fermat's little theorem and the correctness of the RSA scheme, as well as explicit algorithms to perform modular arithmetic. The last chapter provides more graph theory where Eulerian and Hamiltonian cycles are discussed. This book also includes network flows, matchings, covering, bipartite graphs, planar graphs and state the graph minor theorem of Seymour and Robertson.
The book is highly illustrated and each chapter ends with a list of problems of varying difficulty. Undergraduates in mathematics and computer science will find this book useful.
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CHEGG TEXTBOOK SOLUTIONS FOR
Hutchinson s Basic Mathematical Skills With Geometry 8th Edition
Chapter
Problem
PROBLEM
Chapter:Problem:
show all steps
We now present a set of exercises for which the calculator might be the preferred tool. As indicated by the placement of this explanation, you should refrain from using a calculator on the exercises that precede this.
Your scientific or graphing calculator has a key that makes a number negative. This key is different from the minus key that we use for subtraction. The negative key is marked or . With a scientific calculator, the negative key must be pressed after the number is entered. We assume you are using a graphing calculator
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Math modeling handbook now available
Apr 23, 2014
This is the cover of the free math modeling handbook published by the Society for Industrial and Applied Mathematics this month. Credit: SIAM
Math comes in handy for answering questions about a variety of topics, from calculating the cost-effectiveness of fuel sources and determining the best regions to build high-speed rail to predicting the spread of disease and assessing roller coasters on the basis of their "thrill" factor. How does math do all that?
That is the topic of a free handbook published by the Society for Industrial and Applied Mathematics (SIAM) this month: "Math Modeling: Getting Started and Getting Solutions."
Finding a solution to any of the aforementioned problems—or the multitude of other unanswered questions in the real world—will likely involve the creation, application, and refinement of a mathematical model. A math model is a mathematical representation of a real-world situation intended to gain a qualitative or quantitative understanding in order to predict future behavior. Such predictions allow us to come up with novel findings, enable scientific advances, and make informed decisions.
The handbook provides instructions and a process for building mathematical models using a variety of examples to answer wide-ranging questions.
The inspiration for the handbook came from Moody's Mega Math (M3) Challenge, a high school applied math contest organized by SIAM. Despite the tremendous success of the nine-year-old Challenge, which is currently available to 45 U.S. states and Washington, D.C., organizers found that many participating students—high school juniors and seniors—were having trouble coming up with approaches and solutions to the open-ended realistic problems posed by the contest. Participants expressed their frustration in post-contest surveys and emails.
"We have been enthusiastic about the high level of insight and analysis demonstrated by participants in the Challenge, especially the winning teams," says M3 Challenge Project Director Michelle Montgomery. "However, it became clear to us that, given the lack of modeling courses in most high school curricula, many of the participants did not have access to basic resources necessary to create a successful model. We came up with the handbook to give every participant these tools."
This type of thinking created an "aha" moment, so to speak, for handbook authors Karen Bliss, Katie Fowler, and Ben Galluzzo, long-time Challenge judges who have been part of the contest's problem development team for the past two years.
"All students, especially those interested in STEM disciplines, need as much practice in solving open-ended problems as possible, but they often do not get many chances to do that in school,"says Fowler, who is an associate professor of mathematics at Clarkson University. "Math modeling skills allow students to approach problems they initially may feel are outside of their comfort zone, and we want to give them the confidence to tackle them."
Further motivated by a series of SIAM-National Science Foundation (NSF) workshops on the topic of math modeling across the curriculum, the trio began work on a modeling guide. What started as a pamphlet with step-by-step guidance about the modeling process grew into a 70-page, full color handbook, with a companion document that makes connections to the Common Core State Standards as well as easy-to-use reference cards for those who want to get straight to the crux of modeling. The guide is suitable for teachers as well as high school and undergraduate students interested in learning how to model.
"Math modeling is challenging, but it's also surprisingly accessible. The guidebook is designed to remove perceived roadblocks by presenting modeling as a highly-creative iterative process in which multiple approaches—to the same problem—can lead to meaningful results," says Galluzzo, an assistant professor of mathematics at Shippensburg University.
The handbook, as well as the Challenge itself, has another, more pressing goal: motivating our younger generation to pursue higher education and careers in science and math. "SIAM does a big service to the math community at large by giving high school students the opportunity to see how math is more than just a series of formulas and rote memorization," says Bliss, an assistant professor of mathematics at Quinnipiac University. "Students at all levels have the means to produce highly creative solutions to interesting problems. Seeing that math can be a powerful tool for solving truly important problems through M3 Challenge participation might be just enough to encourage a student to study math or another STEM discipline in college."
Related Stories
A paper written by four students from High Technology High School in Lincroft, New Jersey, entitled Ethanol: Not All It Seems To Be has been published in the January 2009 issue of The Mathematical Association of America
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Algebra 2, Noteables: Interactive Study Notebook with Foldables
Algebra 2, Study Guide and Intervention Workbook
Algebra 2, Spanish Study Guide and Intervention Workbook
Algebra 2, Practice Workbook
Algebra 2, Spanish Practice Workbook
Algebra 2, Skills Practice Workbook
Algebra 2, Word Problems Practice Workbook
Algebra 2, Student Edition
Algebra 2, Study Guide & Intervention Workbook
Algebra 2, Homework Practice Workbook
Summary
StudentWorks Plus TM DVD-ROMcombines the complete interactive Student Edition with all of the associated student worksheets in one convenient place. Additionally it includes lesson audio along with Concepts in Motion, Personal Tutor, workbooks and hot links to all the student resources at glencoe.com.
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These courses involved learning the techniques, both analytical and numerical, for solving ordinary, partial, and non-linear differential equations including Green's function, Laplace and Fourier Transform techniques. I have also taught Differential Equation several times at the college. For th...
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booklet containing 31 problem sets that involve a variety of math skills, including scientific notation, simple algebra, and calculus. Each set of problems is contained on one page. Learners will use mathematics to explore varied space...(View More) science topics including black holes, ice on Mercury, a mathematical model of the Sun's interior, sunspots, the heliopause, and coronal mass ejections, among many others.(View Less)
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This is an instruction
system for a second course in Algebra. It is designed for students in grades 8
and above with the knowledge contained in a Beginning Algebra course. The content
includes: graphing on the xy-plane, the use of integer exponents, operations on
polynomials, and solving quadratic equations.
Every objective
is thoroughly explained and developed. Numerous examples illustrate concepts and
procedures. Students are encouraged to work through partial examples. Each unit
ends with an exercise specifically designed to evaluate the extent to which the
objectives have been learned. The student is always informed of any skills that
were not mastered.
The instruction
depends only upon reasonable reading skills and conscientious study habits. With
those skills and attitudes, the student is assured a successful math learning
experience.
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Providing an accessible introduction to the basics of fractals, this DVD presents an appealing balance of the theoretical and aesthetic aspects of the Mandelbrot set. Viewers will appreciate the clarity of exposition as John Hubbard uses a combination of lecture, boardwork, Macintosh computer demonstrations, and colorful computer-generated films and pictures to bring the concepts to life.
Part I focuses on iteration and Julia sets, while Part II addresses Mandelbrot sets. Part III examines a way of using the concept of electric field lines to understand these fractal sets. The concluding remarks round out the lecture by pointing to a philosophical framework that relate these sets to phenomena occurring in the natural world. Requiring only a background in calculus, this is a useful tool in classrooms and an excellent addition to a library.
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You are here
A First Course in Mathematical Analysis
Publisher:
Cambridge University Press
Number of Pages:
459
Price:
50.00
ISBN:
9780521684248
This is truly a student-oriented text on single-variable Advanced Calculus, based on refinements of correspondence texts that have been used by more than 10,000 students in Britain's Open University since 1971.
The first chapter ('Numbers') provides basic background on the real line, including a discussion of inequalities, without introducing any formal topology (compactness, etc.) The book uses a sequential approach to continuity, differentiability, and integration. Null sequences and their properties are introduced and are then used to discuss convergence of general sequences: A sequence {a(n)} converges to L if and only if {a(n) – L} is a null sequence. Continuity of a function f is defined in terms of the behavior of f on convergent sequences. After this kind of gentle preparation involving manipulation of only one epsilon and a (generally large) number X— not necessarily an integer — the student is introduced to the equivalent epsilon-delta definitions of limits and continuity. The epsilon-X and epsilon-delta methods are described as games (a strategy I have used): Player A chooses an epsilon and Player B tries to find an appropriate X or delta…. Proving convergence means that Player B wins. The non-convergence of sequences and functions is also considered.
Four appendices provide useful prerequisite information and solutions to problems (as opposed to 'Exercises', whose answers are not given). In between, most of the standard material of Advanced Calculus is treated in a way that even the 'friendliest' of current texts can only aspire to emulate. There are many helpful diagrams and marginal notes. Although I have often found such marginalia annoying in other texts, I see that in Brannan's book these teaching suggestions, diagrams, and bits of advice to the reader ("You might like to compare this solution with that of Example 3 in Subsection 5.3.2," "You may omit this proof at a first reading,"…) do provide a level of support and comfort to a student who may be reading the text on his or her own or in parallel with a formal course.
Among the gems scattered throughout, we find nicely motivated discussions of π and e, a detailed treatment of the 'Blancmange' function (Takagi fractal curve) — continuous everywhere but differentiable nowhere — a proof of Stirling's formula, some theorems on the location of zeros of polynomials, and a proof that π is irrational. (Although the Index promises a discussion of the irrationality of e on p. 175, there doesn't seem to be a proof anywhere in the book.)
Neither as elegant as Elementary Analysis: The Theory of Calculus by Ross or the recentbooks by Morgan, nor possessing the gravitas of the classic Advanced Calculus books (Apostol, Buck, Kaplan, …), Brannan's book speaks to the average math or science student. The spirit in which this book has been written explains why the Open University was rated the top university in England and Wales for student satisfaction in the last two government national student satisfaction surveys. This text deserves adoption consideration by any instructor who wants to give his or her students a peek behind the curtains of real analysis to see some of its beauty and usefulness revealed.
Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.
Comments
This is an addendum to my published review. Brannan does indeed prove that e is irrational, but on pp. 125-126 (Theorem 2) of the paperbound edition of his text rather than on p. 175. Also, I should have mentioned that the author does not discuss Cauchy sequences at all.
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Read More Equations as they apply to engineering and the sciences. Sound and Accurate Exposition of Theory-special attention is made to methods of solution, analysis, and approximation. Use of technology, illustrations, and problem sets help readers develop an intuitive understanding of the material. Historical footnotes trace development of the discipline and identify outstanding individual contributions.
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Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 97804704583Hardcover. New Condition. SKU: 9780470458310-1Text Book
It's a text book. It meets the requirements of the course.
Starwalk
Apr 28, 2007
First/Second Year Differential Equations
I bought this book because I found I was hogging a copy from the university library. If you are doing differential equations courses in either a mathematics or possibly a physics degree, then I would recommend this as it has in pretty much of anything I would imagine you would need, at least at first and second year level (I'm second year, so cannot give an opinion about use for third/fourth years) There are sections covering both numerical and analytical solutions. Yes, the text can be a bit dry in places, but it never treats you like an idiot, and I don't think many people read books like this for fun.
Looking for extra problems to work through? Buy this book. Lecturer is an idiot and/or lecture notes are poor? Buy
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Mathematical Software deals with software designed for mathematical applications such as Fortran, CADRE, SQUARS, and DESUB. The distribution and sources of mathematical software are discussed, along with number representation and significance monitoring. User-modifiable software and non-standard arithmetic programs are also considered. Comprised of... more...
Introduction to Dynamic Programming provides information pertinent to the fundamental aspects of dynamic programming. This book considers problems that can be quantitatively formulated and deals with mathematical models of situations or phenomena that exists in the real world. Organized into 10 chapters, this book begins with an overview of the fundamental... more...
Presents nearly all the important elementary and analytical methods of statistics, designed for the needs of the geoscientist and completely free from higher mathematics. Translated from the second German edition. more...
This introductory text is geared toward engineers, physicists, and applied mathematicians at the advanced undergraduate and graduate levels. It applies the mathematics of Cartesian and general tensors to physical field theories and demonstrates them chiefly in terms of the theory of fluid mechanics. Numerous exercises appear throughout the text. 1962... more...
This distinctly nonclassical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. The two-part treatment begins with an overview that discusses the theory's historical evolution, followed by a development of related mathematical... more...
This book can be used as either a primary text or a supplemental reference for courses in applied mathematics. Its core chapters are devoted to linear algebra, calculus, and ordinary differential equations. Additional topics include partial differential equations and approximation methods. Each chapter features an ample selection of solved problems.... more...
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The Algebra Survival Guide Workbook
Browse related Subjects ...
Read More like a laser beam on a particular algebra skill, then offers ample practice problems. Answers are conveniently displayed in the back. This book is for parents of schooled students, homeschooling parents and teachers. Parents of schooled children find that the problems give their children a leg up for mastering all skills presented in the classroom. Homeschoolers use the Workbook - in conjunction with the Guide - as a complete Algebra 1 curriculum. Teachers use the workbook's problem sets to help children sharpen specific skills - or they can use the pages as tests or quizzes on specific topics. Like the Algebra Survival Guide, the Workbook is adorned with beautiful art and sports a stylish, teen-friendly design
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state-of-the-art introduction to the powerful mathematical and statistical tools used in the field of finance
The use of mathematical models and numerical techniques is a practice employed by a growing number of applied mathematicians working on applications in finance. Reflecting this development, Numerical Methods in Finance and Economics: A MATLAB®-Based Introduction, Second Edition bridges the gap between financial theory and computational practice while showing readers how to utilize MATLAB®—the powerful numerical computing environment—for financial applications.
The author provides an essential foundation in finance and numerical analysis in addition to background material for students from both engineering and economics perspectives. A wide range of topics is covered, including standard numerical analysis methods, Monte Carlo methods to simulate systems affected by significant uncertainty, and optimization methods to find an optimal set of decisions.
Among this book's most outstanding features is the integration of MATLAB®, which helps students and practitioners solve relevant problems in finance, such as portfolio management and derivatives pricing. This tutorial is useful in connecting theory with practice in the application of classical numerical methods and advanced methods, while illustrating underlying algorithmic concepts in concrete terms.
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More Good Questions, written specifically for secondary mathematics teachers, presents two powerful and universal strategies that teachers can use to differentiate instruction across all math content: Open Questions and Parallel Tasks. Showing teachers how to get started and become expert with these strategies, this book also demonstrates how to use more inclusive learning conversations to promote broader student participation. Strategies and examples are organized around Big Ideas within the National Council of Teachers of Mathematics (NCTM) content strands. With particular emphasis on Algebra, chapters also address Number and Operations, Geometry, Measurement, and Data Analysis and Probability, with examples included for Pre-Calculus.
To help teachers differentiate math instruction with less difficulty and greater success, this resource:
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Not only is the theory sound, but the authors have compiled many many sample problems that are ready to use with a class. They are even organized by the strands used in the Ontario curriculum.
This will certainly be a part of my teaching this year and in the future.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
7 reviews
8 of 9 people found the following review helpful
Not just for high school - Grade 6 and up!Nov. 24 2010
By
SarahQuilts
- Published on Amazon.com
Format: Paperback
I love this book!
I will admit to a slight bias - I met one of the authors, Amy Lin because she worked at my school board and she suggested I buy this book. It is money well spent. I really like that there are many many practical suggestions and example questions that you can use in the classroom right away. The questions are divided by strand, and there are enough that you could use them weekly through an entire year.
I wish that the grade level was indicated in the title. Half of the content is geared towards grades 6-8 (and many of the lower grade questions would be good for high school classes too)
There are two types of activities "open questions" that let students participate in a mathematical discussion at multiple levels, and "parallel tasks" that give student choice, but achieve the same big picture understandings. The authors even included scaffolding questions or prompting questions to help re-start a discussion that has stalled or help a student get started.
Finally, there is a clear index organized by topic that lets you flip straight to the page that has the questions you need for today's lesson.
5 of 6 people found the following review helpful
There are NO questions.March 22 2013
By
Janet Orloski
- Published on Amazon.com
Format: Paperback
Verified Purchase
This book addresses generalities about differentiation and does not include specific lessons that differentiate mathematics. I do not recommend this book.
1 of 1 people found the following review helpful
Great Book!Jan. 8 2013
By
Catwoman
- Published on Amazon.com
Format: Paperback
Verified Purchase
This book has lots of applicable ideas for teaching secondary level mathematics. I read the hints when putting together Lesson Plans. I have two copies one for my personal library and one as a reference in the classroom.
Great Common Core resource!Dec 22 2013
By
Emily Y.
- Published on Amazon.com
Format: Paperback
I have used the questions and parallel tasks for math journals. My students love them because they can think outside the box, and there are no wrong answers. I especially appreciate the sample teacher questions that the authors provide to help facilitate class discussion. I would recommend this book to math teachers looking for a way to incorporate more writing and discussion in their class.
ExcellentOct. 23 2013
By
Freedomite
- Published on Amazon.com
Format: Paperback
Verified Purchase
Very useful in developing good questions to facilitate student centered talk that is needed to support student competencies for the Math Common Core Curriculum.
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Courses
Review of fundamental operations and a more extensive study of fractions, exponents, radicals, linear and quadratic equations, ratio, proportion, variation, progressions, and the binomial theorem. Topics normally included in intermediate algebra in high school. Students who have satisfactorily completed two years of high school algebra, or the equivalent, receive no credit for this course. Offered only in Evening Division and Summer Session. Prerequisite: One year of high school algebra.
Sets, relations, functions with particular attention to properties of algebraic, exponential, logarithmic functions, their graphs and applications in preparation for MATH 019. May not be taken for credit concurrently with, or following receipt of, credit for any mathematics course numbered MATH 019 or above. Pre/co-requisites: Two years of secondary school algebra; one year of secondary school geometry.
Skills in working with numerical, algebraic, and trigonometric expressions are developed in preparation for MATH 021. May not be taken for credit concurrently with, or following receipt of, credit for any mathematics course numbered MATH 019 or above. Prerequisite: Two years of secondary school algebra; one year of secondary school geometry.
Topics include geometry, measurement, probability, statistics, algebra, number theory, and problem solving to provide background for future instruction in elementary and middle school mathematics. Prerequisite: Three years of secondary school math.
Introduction to mathematics of finite systems with applications, such as probability, statistics, graph theory, fair division and apportionment problems, voting systems. Prerequisites: Two years of secondary school algebra or MATH 009 or MATH 010.Introduction to calculus of functions of one variable including: limits, continuity, techniques and applications of differentiation and integration. Prerequisites: MATH 010, or strong background in secondary school algebra and trigonometry. Credit not given for more than one course in the pair MATH 019, MATH 021 unless followed by MATH 022.
Intended to make the transition from a B or better in MATH 019 to MATH 121. Topics are similar to MATH 022 but recognizing different backgrounds of students in MATH 019 versus MATH 021. Prerequisite: B or better in MATH 019. Credit will not be given for both MATH 022 and MATH 023.
Emphasizing proofs, fundamental mathematical concepts and techniques are investigated within the context of number theory and other topics. Co-requisite: MATH 021. Credit not given for both MATH 052 and MATH 054.
Solutions of linear ordinary differential equations, the Laplace transformation, and series solutions of differential equations. Prerequisite: MATH 121. Corequisite: MATH 124 or Instructor permission. Credit not granted for more than one of the courses MATH 230 or MATH 271.
Techniques of Undergraduate calculus and linear algebra are applied for mathematical analysis of models of natural and human-created phenomena. Students are coached to give presentations. Prerequisites: MATH 121 and any of MATH 124, MATH 230, or MATH 271.
Classification of equations, linear equations, first order equations, second order elliptic, parabolic, and hyperbolic equations, uniqueness and existence of solutions. Prerequisite: MATH 230; MATH 242.
Topics will vary each semester and may include algebraic number theory, algebraic geometry, and the arithmetic of elliptic curves. Repeatable for credit with Instructor permission. Prerequisite: MATH 252.
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More About
This Textbook
Overview
In the nineteenth century, French mathematician Evariste Galois developed the Galois theory of groups-one of the most penetrating concepts in modem mathematics. The elements of the theory are clearly presented in this second, revised edition of a volume of lectures delivered by noted mathematician Emil Artin. The book has been edited by Dr. Arthur N. Milgram, who has also supplemented the work with a Section on Applications.
The first section deals with linear algebra, including fields, vector spaces, homogeneous linear equations, determinants, and other topics. A second section considers extension fields, polynomials, algebraic elements, splitting fields, group characters, normal extensions, roots of unity, Noether equations, Jummer's fields, and more.
Dr. Milgram's section on applications discusses solvable groups, permutation groups, solution of equations by radicals, and other 15, 2001
concise and self-contained introduction to galois theory
You don't need any algebra background to read and appreciate this book. Only the knowledge of the definitions of groups and normal subgroups is needed. You can find these in any modern algebra book. I read it as a college sophomore without much prior knowledge in this field. I was able to enjoy it pretty much. It might be a little too dense for beginners, but it is almost entirely self contained. It is written based on lecture notes, so don't expect it to be in a very organized format. The only thing I don't like is that it doesn't have an index, but it's okay since the book is very thin.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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Launchings
Mind the Gap
"Mind the Gap" is an appropriate metaphor for one of the greatest
challenges facing undergraduate mathematics education today. There is a significant
gap between student experience of mathematics in high school and the expectations
they face on entering college, and there are troubling signs that this gap
may be widening. There are serious problems in K-12 mathematics education,
but college faculty also need to look to their own house and think about the
first-year experience of their own students.
In my article "Is the Sky Still Falling?" [1],
I observed that 4-year college mathematics enrollments at the level of calculus
and above declined from 1985 to 1995 and have since recovered to slightly
below the 1990 numbers. Two-year colleges saw calculus enrollments rise in
the early '90s, then fall to well below the 1990 number, while the number
of their students requiring remedial mathematics exactly doubled. In percentages,
the picture is dismal. For 4-year undergraduate programs, calculus and advanced
mathematics enrollments dropped from 10.05% of all students in 1985 to 6.36%
in 2005.
1985
1990
1995
2000
2005
precollege level
251 (3.25%)
261 (3.04%)
222 (2.53%)
219 (2.34%)
201 (1.83%)
introductory level
593 (7.69%)
592 (6.90%)
613 (6.99%)
723 (7.72%)
706 (6.42%)
calculus level
637 (8.26%)
647 (7.54%)
538 (6.14%)
570 (6.09%)
587 (5.34%)
advanced
138 (1.79%)
119 (1.39%)
96 (1.09%)
102 (1.09%)
112 (1.02%)
Table 1. Four1985
1990
1995
2000
2005
precollege level
482 (10.64%)
724 (13.82%)
800 (14.56%)
763 (12.83%)
964 (14.86%)
introductory level
294 (6.49%)
361 (6.89%)
415 (7.56%)
396 (6.66%)
501 (7.72%)
calculus and advanced
113 (2.49%)
140 (2.67%)
142 (2.59%)
117 (1.97%)
120 (1.85%)
Table 2. TwoThis happened while high school students were taking ever more
mathematics at ever higher levels. In 1982, only 44.5% of high school graduates
had completed mathematics at the level of Algebra II or higher. By 2004, this
had risen to 76.7%. In 1982, 10.7% had completed precalculus. By 2004, it
was 33.0%, over a million high school graduates arriving in college ready—at
least in theory—to begin or continue the study of calculus. Yet over
the years 1985–2005, Fall term enrollments in Calculus I dropped from
264,000 to 252,000.
Table 3. Percentage of high school graduates
who completed different levels of mathematics in 1982, 1992, and 2004 [5].
Admittedly, many more students today arrive at college already having earned
credit for Calculus I, but they have not produced larger enrollments for Calculus
II. Over these same twenty years, Fall term enrollments in Calculus II dropped
from 115,000 to 104,000. Across the board, students are arriving in college
and failing to take what should be a next course in their mathematical progression.
The college community is not blameless. Too many good students are turned
off by their initial college experience in mathematics. Too often, first-year
courses are large and impersonal, instructors—especially adjunct faculty
and graduate teaching assistants—are under-prepared, and little thought
has gone into implementing appropriate pedagogies. Moreover, a common complaint
that I hear from high school teachers is that colleges focus exclusively on
what students do not know, with the result that many students find themselves
assigned to classes they find stultifying.
This last is a tricky issue. The answer cannot be that colleges lower their
expectations of what it means to know algebra or calculus. It does mean that
colleges need to rethink how to get students from where they are as they enter
college to where they need to be. It does mean offering more routes into good
mathematics and restructuring existing courses so that they acknowledge and
build upon what students do know while remaining mindful of and addressing
the gaps in this knowledge. Especially when a student needs to relearn a topic
that appears familiar, we must ensure that the course is structured so that
it provides fresh challenges that entice students to keep moving forward.
We have learned a lot about teaching undergraduates in the past twenty years.
There are proven programs for bridging the gap. The Emerging Scholars Program
is one. Stretching Calculus I over two terms with precalculus topics treated
on a just-in-time basis is another. But there are no magic bullets. Each college
and university must examine what others have done and adapt to its own situation
those programs that are most appropriate.
[4] National Center for Education Statistics. 2007.
Digest of Education Statistics. US Department of Education, nces.ed.gov/programs/digest/
[5] National Center for Education Statistics. 2007.
Advanced Mathematics and Science Coursetaking in the Spring High School
Senior Classes of 1982, 1992, and 2004. US Department of Education, nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2007312
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College
in St. Paul, Minnesota, and President of the MAA. You can reach him at bressoud@macalester.edu.
This column does not reflect an official position of the MAA.
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applications of maths in real life ppt
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nctions
algebraic relations among polynomials
Why is Gröbner Bases Theory Attractive
The main problem solved by the theory can be explained in 5 minutes (if one knows operations
addition and multiplication of polynomials).
The algorithm that solves the problem can be learned in 15 minutes
The theorem on which the algorithm is based is nontrivial to invent and to prove.
Many problems in seemingly quite different areas of mathematics can be reduced to the problem of
computing Gröbner bases.
How Can Gröbner Bases Theory is Applied
Given a set F of polynom..................[:=> Show Contents <=:]
se.
The Groebner Basis can help us solve the problem whether the robot can reach a certain point with
center at (a, b, c)
Solution:
In order to find the coordinates of the point we are interested in, we have to find all points the
arm can reach and see if this is one of them
Points that can be reached = points that satisfy the above equations
Solve a system of polynomial equations " find the real roots
The forward kinematic problem
Linear systems " reduced row echelon form
In our case " polynomial equations
Groebner basis " the equivalent
(used to ..................[:=> Show Contents <=:]
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This lesson from Illuminations introduces students to to basic graph theory and Euler circuits. Students will gain hands on experience using graphs. The lesson involves sketching graphs that have Euler paths and...
This lesson from Illuminations asks students to solve a system of linear equations using a practical math problem. The lesson involves question for students; participants are asked to give a short presentation to
This math unit from Illuminations allows students to examine geometric sequences and exponential functions through the example of the placement of frets on stringed instruments. The unit includes two lessons,...
This geometry lesson from Illuminations examines regular polygons. Students will study reflections using hinged mirrors and examine the interior angles of regular polygons. The lesson allows students to look at the...
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Introductory Algebra 2nd edition
0130676829
9780130676825
Details about Introductory Algebra:
For courses in basic algebra, this text presents a student-friendly approach to the subject's main concepts. It offers a text-specific, integrated instructor and student media/print/web support package.
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Rent Introductory Algebra 2nd edition today, or search our site for K. Elayn textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson.
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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).
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for cramming, homework help, and reference
Whether you're cramming, you're studying new material, or you just need a refresher, this compact guide gives you a concise, easy-to-follow review of the most important concepts covered in a typical Algebra I course. Free of review and ramp-up materials, it lets you skip right to the parts where you need the most help. It's that easy!
Set the scene — get the lowdown on everything you'll encounter in algebra, from words andsymbols to decimals and fractions
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Most Mathematics courses have prerequisites that are listed as part of the course description in the Schedule of Classes. Before registering for a Mathematics course, be sure you have completed the stated prerequisite.
Note to all Algebra students:
The Math Department uses a single textbook for the Elementary and Intermediate Algebra sequence. This allows students to complete the Algebra sequence in three different ways: a four semester sequence of MATH 111, 112, 122, and 123 each covering one fourth of the book; a two semester sequence of MATH 110 and MATH 120 each covering half of the book; or a combination of the above. Please see your counselor to be sure you take the correct course.
Do you want to get ready for calculus? You can complete both trigonometry and pre-calculus in just one semester. Get access to additional support and tutoring to help you focus on learning math, and complete both of these courses successfully. Students enroll MATH 130 AG (CRN 43396) now, and we will enroll you in MATH 222 AB (CRN 43558) when classes begin. For more information, please contact Denise Hum at Email: humd@smccd.edu
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Calculus
Quick Links
Webinars
Unit Downloads
Differential Equations
Differential equations are in some sense the "reason for being" of the calculus, with the Fundamental Theorem of Calculus being one solution of the most basic differential equation: given dy/dx = f (x), what is y = F(x)? This unit examines some specific examples of the differential equations (leading to the exponential and logarithmic functions) as well as the important numerical technique of Euler's Method for approximating solutions to differential equations. Other lessons consider some specific differential equations arising in the context of science.
| 677.169 | 1 |
Differential Equations: Computing and Modeling (5TH 15 Edition)
by C. Henry Edwards Publisher Comments
For introductory courses in Differential Equations. This text provides the conceptual development and geometric visualization of a modern differential equations course that is still essential to science and engineering students. It reflects the... (read more)
Damned Lies and Statistics, Updated Edition (12 Edition)
by Joel Best Publisher Comments
Here, by popular demand, is the updated edition to Joel Best's classic guide to understanding how numbers can confuse us. In his new afterword, Best uses examples from recent policy debates to reflect on the challenges to improving statistical literacy... (read more)
Number Theory and Its History (76 Edition)
by Oystein Ore Publisher Comments
"A very valuable addition to any mathematical library." — School Science and Math This book, written by a prominent mathematician and Sterling Professor of Mathematics at Yale, differs from most other books on number theory in two important... (read more)
Basic Linear Partial Differential Equations (07 Edition)
by Francois Treves Publisher Comments
Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every... (read more)
Mathematics for Computer Graphics Applications (2ND 99 Edition)
by Michael E. Mortenson Publisher Comments
Includes new chapters on symmetry, limit and continuity, constructive solid geometry, and the Bezier curve. Provides many new figures and exercises. Contains an annotated suggested reading list with exercises and answers in each chapter. Appeals to both... (read more)
First Course in Differential Equations (2ND 11 Edition)
by J. David Logan Publisher Comments
J. David Logan is Willa Cather Professor of Mathematics at the University of Nebraska Lincoln. His extensive research is in the areas of theoretical ecology, hydrogeology, combustion, mathematical physics, and partial differential equations. He is the... (read more)
Introduction To Algorithms : a Creative Approach (89 Edition)
by Udi Manber Publisher Comments
This book emphasizes the creative aspects of algorithm design by examining steps used in the process of algorithm development. The heart of the creative process lies in an analogy between proving mathematical theorems by induction and designing... (read more)
Mathematical Reflections (97 Edition)
by Peter Hilton Publisher Comments
A relaxed and informal presentation conveying the joy of mathematical discovery and insight. Frequent questions lead readers to see mathematics as an accessible world of thought, where understanding can turn opaque formulae into beautiful and meaningful... (read more)
First Course in Differential Equations (10TH 13 Edition)
by Dennis G. Zill Publisher Comments
A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS, 10th Edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. This proven and accessible book speaks to... (read more)
Discrete and Combinatorial Mathematics (5TH 04 Edition)
by Ralph Grimaldi Publisher Comments
This fifth edition continues to improve on the features that have made it the market leader. The text offers a flexible organization, enabling instructors to adapt the book to their particular courses. The book is both complete and careful, and it... (read more)
Schaum's Outline of Precalculus (Schaum's Outlines)
by Fred Safier Publisher Comments
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Proofs in high school I Students are not taught how to write proofs in high schools today. I The best students will still pick out the basic idea of doing ...
MATHEMATICS FOR NUCLEAR ENGINEERING STUDENTS Besides the descriptions here, see the Department of Mathematics website for more information: lsa.umich.edu ...
GCSE Methods in Mathematics - specification (version 1.2) 2 1 Introduction 1a Why choose AQA? We are the United Kingdomu0027s favourite exam board and more students get their ...
1 Institute for Advanced Study/Park City Mathematics Institute International Seminar: Bridging Policy and Practice in the Context of Reasoning and Proof 3-8 July 2006 Problem ...
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Learning Math: Patterns, Functions and Algebra
Linear Functions and Slope (#105)
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations. [27 "LEARNING MATH: Patterns, Functions, and Algebra," organized around NCTM content standards, strives to help teachers better understand the mathematics concepts underlying the content that they teach. This professional development series explores the 'big ideas' in algebraic thinking, such as finding, describing, and using patterns; using functions to make predictions; understanding linearity and proportional reasoning; understanding non-linear functions; and understanding and exploring algebraic structure.
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0078296Mathematics: Applications and Concepts, Course 3, Student Edition
Mathematics: Applications and Concepts is a three-text Middle School series intended to bridge the gap from Elementary Mathematics to High School Mathematics. The program is designed to motivate middle school students, enable them to see the usefulness of
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Elementary and Intermediate Algebra - 3rd edition high
light the relevance of what you are learning. And studying is easier than ever with the book's multimedia learning resources, including ThomsonNOW for ELEMENTARY AND INTERMEDIATE ALGEBRA, a personalized online learning companion.'s multimedia learning resources, including ThomsonNOW for ELEMENTARY AND INTERMEDIATE ALGEBRA, a personalized online learning companion. ...show less
4 EXPONENTS AND POLYNOMIALS. 4.1 Multiplication with Exponents. 4.2 Division with Exponents. 4.3 Operations with Monomials. 4.4 Addition and Subtraction of Polynomials. 4.5 Multiplication with Polynomials. 4.6 Binomial Squares and Other Special Products. 4.7 Dividing a Polynomial by a Monomial. 4.8 Dividing a Polynomial by a Polynomial. Summary. Review/Test. Projects.
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Unit by pengxuebo
Unit 6: Modelling with More Than One Function MHF4U
Lesson Outline
Big Picture
Students will:
consolidate understanding of characteristics of functions (polynomial, rational, trigonometric, exponential,
and logarithmic);
create new functions by adding, subtracting, multiplying, dividing, or composing functions;
reason to determine key properties of combined functions;
solve problems by modelling and reasoning with an appropriate function (polynomial, rational, trigonometric,
exponential and logarithmic) or a combination of those functions.
Day Lesson Title Math Learning Goals Expectations
1 Under Pressure Solve problems involving functions including those from D3.1, 3.3
real-world applications.
GSP® file: Reason with functions to model data. CGE 2b
Under Pressure
Reflect on quality of ‗fit' of a function to data.
2 Solving Inequalities Understand that graphical and numerical techniques are D3.1, 3.2, 3.3
needed to solve equations and inequalities not accessible by
Presentation file: standard algebraic techniques. C4.1, 4.2, 4.3
Inequalities Make connections between contextual situations and
information dealing with inequalities.
Reason about inequalities that stem from contextual
situations using technology.
3 Growing Up Soy Model data by selecting appropriate functions for particular D3.1, 3.3
Fast! domains.
Solve problems involving functions including those from CGE 5a
GSP® file: real-world applications.
The Chipmunk
Reason with functions to model data.
Problem
Reflect on quality of ‗fit' of phenomena to functions that
have been formed using more than one function over the
domain intervals.
4 Combining Make connections between the key features of functions to D2.1, 2.2, 2.3, 3.1
Functions Through features of functions created by their sum or difference
Addition and (i.e., domain, range, maximum, minimum, number of zeros, CGE 4b, 5g
Subtraction odd or even, increasing/decreasing behaviours, and
instantaneous rates of change at a point).
Make connections between numeric, algebraic and graphical
representations of functions that have been created by
addition or subtraction.
Reason about the connections made between functions and
their sums or differences.
5 Combining Connect key features of two given functions to features of D2.1, 2.3, 3.1
Functions Through the function created by their product.
Multiplication Represent functions combined by multiplication numerically, CGE 4b, 5g
algebraically, and graphically, and make connections
between these representations.
Determine the following properties of the resulting
functions: domain, range, maximum, minimum, number of
zeros, odd or even, increasing/decreasing behaviours, and
instantaneous rates of change at a point.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 1
Day Lesson Title Math Learning Goals Expectations
6 Combining Connect key features of two given functions to features of D2.1, 2.3, 3.1
Functions Through the function created by their quotient.
Division Represent functions combined by division numerically, CGE 4b, 5g
algebraically, and graphically, and make connections
Presentation file: between these representations.
Asymptote Becomes
Determine the following properties of the resulting
A Hole
functions: domain, range, maximum, minimum, number of
zeros, odd or even, increasing/decreasing behaviours, and
instantaneous rates of change at a point.
7 Composition of Determine the composition of functions numerically and D2.4, 2.7
Functions graphically.
Numerically and Connect transformations of functions with composition of CGE 4f
Graphically functions.
Explore the composition of a function with its inverse
numerically and graphically, and demonstrate that the result
maps the input onto itself.
8 Composition of Determine the composition of functions algebraically and D2.5, 2.7
Functions state the domain and range of the composition.
Algebraically Connect numeric graphical and algebraic representations. CGE 4f
Explore the composition of a function with its inverse
algebraically.
9 Solving Problems Connect transformations of functions with composition of D2.5, 2.6, 2.8
Involving functions.
Composition of Solve problems involving composition of two functions CGE 4f
Functions including those from real-world applications.
Reason about the nature of functions resulting from the
Winplot file:
composition of functions as even, odd, or neither.
U6L7_8_9.wp2
10 Putting It All Make connections between key features of graphs D3.1
Together (Part 1) (e.g., odd/even or neither, zeros, maximums/minimums,
positive/negative, fraction less than 1 in size) that will have CGE 4f
an affect when combining two functions from different
families.
Identify the domain intervals necessary to describe the full
behaviour of a combined function.
Graph a combined function by reasoning about the
implication of the key features of two functions.
Understand graphs of combined function by reasoning about
the implication of the key features of two functions, and
make connections between transformations and composition.
11– Putting It Altogether Consolidate applications of functions by modelling with Overall D2, D3
12 (Part 2) more than one function.
Consolidate procedural knowledge when combining CGE 2b, 2c, 3c,
functions. 5g
Communicate about functions algebraically, graphically, and
orally
Model real-life data by connecting to the various
characteristics of functions.
Solve problems by modelling and reasoning.
13 Jazz Day
14 Summative
Assessment
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 2
Unit 6: Day 1: Under Pressure MHF4U
Math Learning Goals Materials
Solve problems including those from real-world applications. BLM 6.1.1
computers with
Reason with functions to model data.
Reflect on quality of ‗fit' of a function to data.
GSP® software
75 min
Assessment
Opportunities
Minds On… Whole Class Discussion
Introduce the lesson using the context of a leaky tire and discuss why it is
important to know tire pressure when driving.
Tire pressure is a measure of the amount of air in your vehicle's tires, in pounds
per square inch or kPa (1 psi = 6.89 kPa). If tire pressure is too high, then less of m/content/advice/tire
the tire touches the ground. As a consequence, your car will bounce around on the pressure.html
road. When your tires are bouncing instead of being firmly planted on the road,
you have less traction and your stopping distance increases. If tire pressure is too g/wiki/Pressure
low, then too much of the tire's surface area touches the ground, which increases
friction between the road and the tire. As a result, not only do your tires wear
prematurely, but they also could overheat. Overheating can lead to tread separation
— and a serious accident.
Think/Pair/Share Discussion
Individually, students use the data in to hypothesize a graphical model
(BLM 6.1.1 Part A). Student pairs sketch a possible graph of this relationship.
Invite pairs to share their predictions with the entire class.
Lead a discussion about the meaning of ‗tolerance' in the context of ―hitting‖ a
point on the curve.
Some people suggest that traditional two-sided tolerances are analogous to ―goal
posts‖ in a football game: This implies that all data within those tolerances are
equally acceptable. The alternative is that the best product has a measurement g/wiki/Tolerance_(en
which is precisely on target. gineering)
Action! Pairs Investigation
Students use BLM 6.1.1 Parts B and C and the GSP® file to manipulate each
function model using sliders. They determine which model – linear, quadratic, or Under Pressure.gsp
exponential – best fits the data provided and form an equation that best fits the
data.
Students discuss with their partner other factors that would limit the
appropriateness of each model in terms of the context and record their answers
(BLM 6.1.1 – Part C). Circulate and assist students who may have difficulty
working with the GSP® sketch.
Reasoning/Observation/Mental Note: Observe students facility with the
inquiry process to determine their preparedness for the homework assignment.
Consolidate Whole Class Discussion
Debrief Students present their models for the tire pressure/time relationship and
determine which pair found the ―best‖ model for the data. This could be done
using GSP® or an interactive whiteboard. Discuss the appropriateness of each
model in this context, including the need to limit the domain of the function.
Curriculum Expectations/BLM/Anecdotal Feedback: Provide feedback on
student responses (BLM 6.1.1).
Home Activity or Further Classroom Consolidation (Possible Answer:
Exploration Complete the follow up questions in Part C, if needed, and Part D ―Pumped Up‖
Application on Worksheet 6.1.1. p 7 x 14
52 pumps)
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 3
6.1.1: Under Pressure
Part A – Forming a Hypothesis
A tire is inflated to 400 kilopascals (kPa) and over a few hours it goes down until the tire is quite
flat. The following data is collected over the first 45 minutes.
Pressure, P,
Time, t, (min)
(kPa)
0 400
5 335
10 295
15 255
20 225
25 195
30 170
35 150
40 135
45 115
1 psi = 6.89 kPa
Create a scatterplot for P against time t. Sketch the curve of best fit for tire pressure.
Part B – Testing Your Hypothesis and Choosing a Best Fit Model
The data is plotted on The Geometer's Sketchpad® in a file called Under Pressure.gsp.
Open this sketch and follow the Instructions on the screen.
Enter your best fit equations, number of hits, and tolerances in the table below.
Linear Model Quadratic Model Exponential Model
f x mx b f x a x h k
2
f x a b xh k
Your Best Fit Equations
Number of Hits
Equation of the best fit model:
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 4
6.1.1: Under Pressure (continued)
Part C – Evaluating Your Model
1. Is the quadratic model a valid choice if you consider the entire domain of the quadratic
function and the long term trend of the data in this context? Explain why or why not.
2. Using each of the 3 "best fit" models, predict the pressure remaining in the tire after 1 hour.
How do your predictions compare? Which of the 3 models gives the most reasonable
prediction? Justify your answer.
3. Using each of the 3 "best fit" models, determine how long it will take before the tire pressure
drops below 23 kPA? (Note: The vehicle in question becomes undriveable at that point.)
4. Justify, in detail, why you think the model you obtained is the best model for the data in this
scenario. Consider more than the number of hits in your answer.
Part D: Pumped Up
Johanna is pumping up her bicycle tire and monitoring the pressure every 5 pumps of the air
pump. Her data is shown below. Determine the algebraic model that best represents this data
and use your model to determine how many pumps it will take to inflate the tire to the
recommended pressure of 65 psi.
Number of Pumps Tire Pressure (psi)
0 14 1 psi = 6.89 kPa
5 30
10 36
15 41
20 46
25 49
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 5
Unit 6: Day 2: Solving Inequalities MHF4U
Math Learning Goals Materials
Understand that graphical and numerical techniques are needed to solve equations BLM 6.2.1 or
and inequalities not accessible by standard algebraic techniques. graphing
Make connections between contextual situations and information dealing with
technology
graph paper or
inequalities. graphing
Reason about inequalities that stem from contextual situations using technology. technology
computer and
data projector for
75 min presentation
Assessment
Opportunities
Minds On… Think/Pair/Share Whole Class Discussion
Individually students identify three ways to solve the equation
Inequalities.ppt.
x 3 5x 3; then share with a partner.
See the notes in the
Debrief strategies as a whole class.
first slide.
Lead a discussion about the numerical, algebraic and graphical methods of
solving this problem, using the first six slides of the Inequalities presentation to Students could use
graphing technology
visually demonstrate the graphical solution. to sketch the graph of
the equation, and
Action! Pairs Investigation Whole Class Discussion then they use
reasoning to identify
With a partner, students investigate three ways to solve x 3 5x 3. the solution to the
Lead a discussion about the numerical, algebraic and graphical methods of inequality from the
graph.
solving this problem, using slide 7 to visually demonstrate the graphical solution.
Lead a discussion on the graphical solution of x2 7 x 1 using slides 8 and 9. Alternative
Approach
In pairs, students investigate the graphical solution to x2 7 x 1 and Divide class into
x2 x 6 0. groups; each group
investigates a
Lead a discussion of the solution using slides 10 and 11. different inequality.
Each group presents
Reasoning/Observation/Mental Note: Observe students' reasoning to solve the to the class. Debrief
inequality once the graph is established. by showing the
presentation.
Repeat the pairs investigation, discussion, using the graphs of
x
1 1 3
5, x 2 sin x , and .
x 1
3 x3 9 x 2
Consolidate Whole Class Discussion
Debrief Emphasize the value of multiple representations in the light of some inequalities
being unsolvable without the graphical representation.
Provide a contextual problem:
$1000 is invested at 5% compounded annually. $750 is invested at 7%
compounded annually. When will the $750 investment amount surpass the $1000
investment amount?
Students express this question algebraically (Answer: 1000 1.05 x 750 1.07 x . )
Demonstrate how easy it is to solve graphically by displaying the graph
(BLM 6.2.1).
Home Activity or Further Classroom Consolidation
Solve the inequalities involving quadratics and cubics both algebraically and
Consolidation graphically.
Application Solve the some inequalities involving rational, logarithmic, exponential and
trigonometric functions graphically, using technology, as needed. Determine
some contexts in which solving an inequality would be required.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 6
6.2.1: Solution to CONSOLIDATE Problem
An investment of $750 will exceed an investment of $1000 in about 15.25 years.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 7
Unit 6: Day 3: Growing Up Soy Fast! MHF4U
Math Learning Goals Materials
Model data by selecting appropriate functions for particular domains. BLM 6.3.1, 6.3.2,
Solve problems involving functions including those from real-world applications. 6.3.3, 6.3.4, 6.3.5
graphing
Reason with functions to model data.
calculators
Reflect on quality of ‗fit' of phenomena to functions that have been formed using
computer with
more than one function over the domain intervals. GSP® software
75 min
Assessment
Opportunities
Minds On… Individual Exploration
Students hypothesize about the effects of limiting fertilizer on the growth of the The initial discussion
of the different
Glycine Max (commonly known as the soybean plant), under the three given models students
conditions (BLM 6.3.1). They sketch their predictions and rationales (5 minutes). think would be
appropriate is
Small Group Discussion important to help
Students discuss their choice of model and share their reasoning. They can them properly
change their models after reflection. connect the context
to the mathematical
characteristics of the
functions they have
been studying.
Action! Pairs Investigation
Students use their knowledge of function properties and the data (BLM 6.3.2) to Modelling with
determine function models for each scenario in the experiment. functions becomes
more relevant when
(See BLM 6.3.4.) students recognize
Scenarios two and three provide an opportunity to model relationships by that rarely does a
separating the domain into intervals, and by using different functions to model single function serve
as an appropriate
the data for each interval. model for a real-
Students reflect back to their original predictions (BLM 6.3.1). world problem.
Reasoning/Observation/Mental Note: Listen to students' reasoning for
appropriate function selections and domain intervals to identify student Other relationships
involving more than
misconceptions. one function can be
found at the E-STAT
website
Consolidate Pairs/Whole Group Discussion
Debrief Sample files:
Identify pairs to present their models to the class. Presenters justify their v737344, v151537
reasoning by responding to questions. and v130106.
Discuss the ―fit‖ to their original predictions.
Home Activity or Further Classroom Consolidation Solution provided in
Consolidation Using graphing technology, determine a model that could describe the given Chipmunk
Application relationship by separating the domain into intervals and by using different Problem.gsp
functions for each interval (Worksheet 6.3.3).
See BLM 6.3.5.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 8
6.3.1: Growing Up Soy Fast!
Your biology class is studying the lifecycle of Glycine max (the plant more commonly known as
soybean). You will investigate the effects of limiting the amount of food (fertilizer) used for the
plants' growth.
Group A fertilizes its plants regularly for the first week, does not give any fertilizer for the
2nd week, and then returns to the regular amounts of fertilizer for the 3rd week of the study.
Group B feed its plants regularly for the first week, and then a regularly increased amount of
fertilizer until the end of the study.
Group C slowly increases the amount of fertilizer for the first 10 days, then feeds its plants
regularly for the remainder of the study.
Make predictions about the relationship between each day (from beginning of study) and the
plant height (cm) for each of the groups. Sketch your predictions below and explain your
reasoning.
Group A
Sketch Rationale
Group B
Sketch Rationale
Group C
Sketch Rationale
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 9
6.3.2: Growing Up Soy Fast!
The heights of the plants were measured throughout the study and the following data was taken
by each group:
Scenario A Scenario B Scenario C
Day Height (cm) Day Height (cm) Day Height (cm)
1 3.7 1 3.5 1 4.4
2 9.5 3 6 3 5.8
5 24.0 5 8.5 4 6.7
6 26.4 7 10 7 10.4
8 28.5 9 14.1 8 12.0
11 30.1 11 18.2 10 15.9
14 33.1 12 22.3 13 31
16 37.7 15 26.4 15 40.9
18 45.6 17 30.5 16 46
20 58.2 20 34.6 19 61.1
21 66.4 21 38.7 21 71
Analysing the Data for Scenario A
1. Use your graphing calculator to construct a scatterplot for the Scenario A data. Sketch the
scatterplot you obtained and label your axes.
Height
Day
2. Perform an analysis of the data and, selecting from the functions you have studied, identify
the type of function that you think best models it.
3. Use your knowledge of function properties to determine a function model that best fits the
data.
My function model is:
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 10
6.3.2: Growing Up Soy Fast! (continued)
Analysing the Data for Scenario B
4. Enter the data for Scenario B into the graphing calculator and construct a scatterplot for the
data. Complete both of the screen captures below using the information from your
calculator.
Height
Day
5. A member working with Scenario B decided to use intervals of two different functions to fit
the data where the first function was used to model the first seven days and then the second
function to model the next 14 days. Determine the two functions you feel best fits the data
for the domain intervals identified.
First Function Model: ______________________ x | 0 x 7
Second Function Model: _____________________ x | 7 x 21
Analysing the Data for Scenario C
6. Enter the data for Scenario C into the graphing calculator and construct a scatterplot for the
data. Complete both of the screen captures below using the information from your
calculator.
Height
Day
7. Determine a function model(s) to best fit the data. If using different function models, identify
the domain interval appropriate for each function. Justify your reasoning for your choice of
model(s).
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 11
6.3.3: The Chipmunk Explosion
Chipmunk Provincial Park has a population of about 1000 chipmunks. The population is growing
too rapidly due to campers feeding them. To curb the explosive population growth, the park
rangers decided to introduce a number of foxes (a natural predator of chipmunks) into the park.
After a period of time, the chipmunk population peaked and began to decline rapidly.
The following data gives the chipmunk population over a period of 14 months.
Time Population
(months) (1000s)
1 1.410
2 1.970
3 2.690
5 5.100
6 5.920
7 5.890
9 4.070
9.5 3.650
10 3.260
11 2.600
12 2.090
13 1.670
14 1.330
Use graphing technology to create a scatter plot of the data.
1. Determine a mathematical function model that represents this data. It may be necessary to
use more than one type of function. Include the domain interval over which each type of
function applies to the model.
2. Determine when the population reaches a maximum and what the maximum population is.
3. Determine when the population will fall to less than 100 chipmunks.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 12
6.3.4: Solutions
Analysing the Data for Scenario A
1.
Height
Day
2. The type of function is cubic.
3. My function formula is: y 0.035 x 10 30.
3
Analysing the Data for Scenario B
4.
Height Height
Day Day
5. First Function Model: y 2.5 x 1 x | 0 x 7
,
Second Function Model: y 2.1x 9.8, x | 7 x 21
Note: On a graphing calculator this would be entered as:
y 2.5 x 1 x 0 x 7 2.1x 9.8 x 7 x 21
Height
Day
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 13
6.3.4: Solutions (continued)
Analysing the Data for Scenario C
6.
Height
Day
y 3.8 1.15 , x | 0 x 10 and y 5 x 34 , x | 10 x 21 .
x
7.
Note: On a graphing calculator this would be entered as:
y 3.8 1.15
x
x 0 x 10 5 x 34 x 10 x 21
Height
Day
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 14
6.3.5: Solutions – The Chipmunk Problem
Using the Graphing Calculator
Stat Plot Functions
Height
Day
1. Stat Plot with Functions:
Height
Day
2. Maximum Population:
Height
Maximum population is 6000 at 6.5 months. Day
3. Determination of when population reaches 500:
Height Population reaches 500 at about 18.4 months.
Day
Height
Day
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 15
6.3.5: Solutions – The Chipmunk Problem (continued)
Using GSP®
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 16
Unit 6: Day 4: Combining Function Through Addition and Subtraction MHF4U
Math Learning Goals Materials
Make connections between the key features of functions to features of functions BLM 6.4.1–6.4.5
graphing
created by their sum or difference (i.e., domain, range, maximum, minimum,
number of zeros, odd or even, increasing/decreasing behaviours, and instantaneous calculators
rates of change at a point).
Make connections between numeric, algebraic and graphical representations of
functions that have been created by addition or subtraction.
Reason about connections made between functions and their sums or differences.
75 min
Assessment
Opportunities
Minds On… Whole Class Demonstration
Note: A polynomial
Demonstrate the motion a child on a swing by swinging a long pendulum in front
of degree 4 that is
of a CBR. Students anticipate the graph of the distance of the pendulum from the even could be
CBR over 15 seconds. They use the CBR to capture the graph of the pendulum included to enrich the
motion. (Note: It will be a damped sinusoidal function.) activities.
Students compare their anticipated graph with the actual graph. They graph its Some pairs will have
motion if it were to continue and tell what function represents this motion. identical
combinations.
Lead them to the understanding that no one function would ―work,‖ but 2 Additional
different functions could be combined to produce this particular graph. combinations can be
(Answer: A sin function divided by an exponential function). See Day 6 for generated and used.
further investigation.
Groups of 3 or 4 Activity Possible Check
Distribute one function from BLM 6.4.1, to each group and the blank template Assignments
(BLM 6.4.2). Bring to their attention that the graphing window used for each F 1 checks 5 and 6
F 2 checks 3 and 7
function was: 5 x 5 and 10 y 10 . The functions used: F 3 checks 2 and 4
F 4 checks 1 and 6
Function 1: y = –(x – 1)2 + 3 Function 2 : y = 2x – 1 F 5 checks 3 and 7
F 6 checks 2 and 5
1
x
Function 3: y = (x + 2)(x – 1)x Function 4: y = 2 F 7 checks 1 and 4
Function 5: y log3 x Function 6: y 5cos x
Alternate
Function 7: y = 1
x 2 Questioning
If this is the offspring
function, which of
Students identify the type of function and its key features and properties. They these functions might
post their function and its properties (BLM 6.4.2). be its parents?
Assign each group to check the work of two other groups. Students add to or
correct as they check the two assigned functions.
Combinations
Pairs Anticipation Offspring 1 = F1 + F2
Offspring 2 = F6 + F7
Distribute one function (BLM 6.4.3) to each pair of students and BLM 6.4.4. Offspring 3 = F2 + F3
Each function is a combination of the ones posted. Students look at their Offspring 4 = F5 + F4
combination and predict which two combined to produce it. Offspring 5 = F2 + F4
Pairs share their offspring and the two functions they think combined to produce
their offspring giving reasons for their response.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 17
Unit 6: Day 4: Sum Kind of Function (continued) MHF4U
Assessment
Opportunities
Action! Pairs Investigation
Pairs investigate the addition of two functions in detail to connect the algebraic, Alternate
investigation
numeric, and graphical representations (BLM 6.4.5). They share their results and combining functions
make generalizations, if possible. As Pair A, B, C, or D students determine
which pair of functions they will investigate. wolfram.com/Combini
ngFunctions/
Circulate, address questions, and redirect, as needed. Identify pairs of students to
present their results and generalization.
Reasoning/Observation/Mental Note: Observe students facility with the
inquiry process to determine their preparedness for the homework assignment.
Consolidate Whole Class Discussion
Debrief Pairs present their solution to the class.
Lead a discussion to make conclusions about the connection between the
algebraic, graphical, and numeric representations of the sums of functions.
Discuss key properties and features of their sum; how they relate to the original
functions; strategies used that were useful; and any misconceptions.
Home Activity or Further Classroom Consolidation
Exploration Examine the differences in your pair of functions. Assign a different
Application pair of functions to
each student.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 18
BLM 6.4.1: The Mamas and the Papas
Function 1 Function 2
F1 x F2 x
Function 3 Function 4
F3 x F4 x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function May 11, 2011 19
BLM 6.4.1: The Mamas and the Papas (continued)
Function 5 Function 7
F5 x F7 x
Function 6
F6 x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function May 11, 2011 20
BLM 6.4.2: The Key Features of the Parents
Fill in the chart for the function your group has been assigned. Post the function and chart.
Function Type:
Zeros:
Maxima/Minima:
Asymptotes:
Domain:
Range:
Increasing/Decreasing Intervals:
General Motion of Curve:
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function May 11, 2011 21
BLM 6.4.3: The Offspring Functions
Offspring 1 Offspring 3
Offspring 2
Offspring 5
Offspring 4
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function May 11, 2011 22
BLM 6.4.4: Who Made Who?
Offspring ______
What type of function does it look like?
If this function is a combination of two of the functions posted around the room, which two might
it be? Which one can it not be? Give reasons.
Consider the key features of the function and the ones that you think combined to make it, what
is similar between them?
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function May 11, 2011 23
BLM 6.4.5: Investigating Addition
1
x –2 –1 0 1 2 3
2
a) F1 x
b) F2 x
c) F1 x F2 x
2. a) Plot the points for F1 x .
Sketch and label the graph F1 x .
b) Similarly, sketch and label the graphs
of F2 x .
c) Use your table values and reasoning to
sketch and label the graph
of F x F1 x F2 x .
3. Determine F x F1 x F2 T F1 1 , m T F2 1 , and mT F 1 . How do these
values compare?
7. Compare your results with another group who added the same pair of functions.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function May 11, 2011 24
BLM 6.4.5: Investigating Addition of Functions (continued)
1
Pair B F6 ( x) 5cos x, F7 ( x)
( x 2)
1. Fill in the following table of values
x –4 –2.5 –2 –1 0 1 π 4
2 2
a) F6 x
b) F7 x
c) F6 x F7 x
2. a) Plot the points for F6 x .
Sketch and label the graph F6 x .
b) Similarly, sketch and label the graphs of F7 x .
c) Use your table values and reasoning to sketch
and label the graph of F x F6 x F7 x .
3. Determine F x F6 x F76 2 , mT F7 2 and mT F 2 � 25
BLM 6.4.5: Investigating Addition of Functions (continued)
Pair C F2 x 2 x 1, F3 x x x 2 x 1
1. Fill in the following table of values
1
x –3 –2 –1 0 1 2 3
2
a) F2 ( x)
b) F3 ( x)
c) F2 ( x) F3 ( x)
2. a) Plot the points for F2 ( x) .
Sketch and label the graph F2 ( x) .
b) Similarly, sketch and label the graphs of F3 ( x) .
c) Use your table values and reasoning to sketch
and label the graph of F ( x) F2 ( x) F3 ( x) .
3. Determine F ( x) F2 ( x) F32 1 , mT F3 26
BLM 6.4.5: Investigating Addition of Functions (continued)
x
1
Pair D F4 ( x ) , F5 ( x) log3 x
2
1. Fill in the following table of values
x –4 –3 –2 –1 0 1 2 3 4
a) F4 ( x)
b) F5 ( x)
c) F4 ( x) F5 ( x)
2. a) Plot the points for F4 ( x) .
Sketch and label the graph F4 ( x) .
b) Similarly, sketch and label the graphs of F5 ( x) .
c) Use your table values and reasoning to sketch
and label the graph of F ( x) F4 ( x) F5 ( x) .
3. Determine F ( x) F4 ( x) F54 1 , mT F5 27
Unit 6: Day 5: Combining Functions Through Multiplication MHF4U
Math Learning Goals Materials
Connect key features of two given functions to features of the function created by BLM 6.5.1
their product.
Represent functions combined by multiplication numerically, algebraically, and
graphically, and Small Group Discussion
Students compare solutions with others who worked on the same two difference
functions from during the Home Activity.
Discuss as a class key properties and strategies for a difference of functions.
Compare to sum of functions from Day 4.
(Possible Observations: When subtracting two functions, the x-intercept is the
intersection point of the original two functions.) Answer
1. Always; since
Students reason if each statement is always, sometimes, or never true, and justify
f x 0 at this
their answer using examples and/or reasoning that can be described with the help
point, the sum
of a graph.
f x g x g x
1. When adding f x and g x , at the x-intercept of f x , the sum will (always,
2. Sometimes; only
sometimes, never) be a point on the graph of g x . true when the
2. When adding f x and g x , at the place where f x and g x intersect the intersection of
f x and
sum will (always, sometimes, never) be a point on f x g x .
g x occurs on the
3. f x g x (always, sometimes, never) equals g x f x .
x-axis; otherwise
not true.
Action! Pairs Investigation
3. Always; addition
Student pairs numbered A, B, C, or D, determine for which pair of functions they is commutative.
will investigate products (BLM 6.5.1). Students compare work with another pair
who has worked on the same set of functions.
The functions will be
Circulate to identify pairs to present their solutions and address questions.
selected from F1
Reasoning/Observation/Mental Note: Observe students' facility with the through F6 as per
inquiry process to determine their preparedness for the homework assignment. Day 4.
Consolidate Whole Class Discussion
Debrief Identified pairs present their work to the class.
In a teacher-led discussion, make some conclusions about the connection
between the algebraic, graphical, and numeric representations of the quotient of
functions.
Discuss key properties and features of the product, and how they relate to the
original functions.
Home Activity or Further Classroom Consolidation
1. Journal Entry
When subtracting two functions, what is the significance of the
intersection point of the graphs?
Consolidation Compare the significant points and characteristics to consider when
Application graphing the sum and difference of functions. Which are the same? Which
are different? Explain.
Summarize the important points and intervals to consider when Assign two new
multiplying functions. functions.
2. Graph the product of the new functions.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 28
6.5.1: Investigating the Product of Functions
Pair A F1( x) ( x 1)2 3, F2 ( x) 2 x 1
1. Fill in the following table of values
1
x –2 –1 0 1 2 3
2
a) F1( x)
b) F2 ( x)
c) F1( x) F2 ( x) sketch
and label the graph of F ( x) F1( x) F2 ( x) .
3. Determine F ( x) F1( x) F2 ( x) multiplied the same pair of functions.
TIPS4RM: MHF4U: Unit 6: Modelling with More Than One Function 2008 29
6.5.1: Investigating the Product of Functions (continued)
1
Pair B F6 ( x) 5cos x, F7 ( x )
x2
1. Fill in the following table of values
x –4 –2.5 –2 –1 0 1 4
2 2
a) F6 ( x)
b) F7 ( x)
c) F6 ( x) F7 ( x)
2. a) Plot the points for F6 ( x) .
Sketch and label the graph F6 ( x)
b) Similarly, sketch and label the graphs of F7 ( x) .
c) Use your table values and reasoning to sketch
and label the graph of F ( x) F6 ( x) F7 ( x) .
3. Determine F ( x) F6 ( x) F7 ( x) algebraically.6 2 , mT F7 2 , and mT F 2 30
6.5.1: Investigating the Product.c) F2 x F66 x .
3. Determine F x F2 x F6 x 1 , mT F3 1 and mT F 1� 31
6.5.1: Investigating the Product of Functions (continued)
Pair D F2 x 2 x 1, F5 x log3 x
1. Fill in the following table of values.
1
x –3 –1 0 1 3 9
2
a) F2 x
b) F5 x
c) F2 x F555 x .
3. Determine F x F2 x F5 x algebraically.
Verify 3 of your results from c) numer 32
Unit 6: Day 6: Combining Functions Through Division MHF4U
Math Learning Goals Materials
Connect key features of two given functions to features of the function created by BLM 6.6.1
computer and
their quotient.
Represent functions combined by division numerically, algebraically, graphically,
data projector for
presentation
and Whole Class Discussion
Discuss what is occurring in each of these situations by considering and
reflecting on the numeric, graphical, and algebraic representations.
Ask: For which values of x will the following functions result in:
a) a positive b) negative c) a very small number
d) a very large number e) result is 0 f) undefined
x2 1 x2 1
(i) 2 2
(ii) sin x
x (iii) x2 (iv) x 1
Asymptote
becomes a hole.ppt
Emphasize the difference between an asymptote and a ―hole‖ in the graph.
Action! Pairs Investigation
Student pairs numbered as A, B, C, or D, (BLM 6.6.1) determine for which pair Functions should be
of functions they will investigate quotients. Students compare work with other different from the
pairs who have worked on same set of functions. Circulate to address questions previous day's
assignment.
and identify pairs to present their solutions on the overhead.
Reasoning/Observation/Mental Note: Observe students' facility with the
inquiry process to determine their preparedness for the homework assignment.
Consolidate Pairs Whole Class Discussion
Debrief Identified pairs present their work to the class. Lead a discussion to make some
conclusions about the connection between the algebraic, graphical, and numeric
representations of the quotient of functions. Discuss key properties and features
of the product, and how they relate to the original functions.
Home Activity or Further Classroom Consolidation
Journal Entry: Assign students two
Compare the significant points and characteristics to consider when graphing functions from which
Consolidation
the product and quotient of functions. they will graph the
Application quotient.
Which are the same? Which are different? Explain.
Summarize the important points and intervals to consider when multiplying or
dividing functions.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 33
6.6.1: Investigating the Division.
1
x –2 –1 0 1 2 3
2
a) F1( x)
b) F2 ( x)
F1( x )
c)
F2 ( x )
F1( x)
sketch and label the graph of F x .
F2 ( x)
F1( x)
3. Determine F x algebraically.
F2 divided the same pair of functions.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 34
6.6.1: Investigating the Division of Functions (continued)
Pair B F6 x 5 cos x , F2 x 2 x 1
1. Fill in the following table of values
1
x –4 –2 –1 0 1 1 4
2 2
a) F6 ( x)
b) F2 x
F6 ( x )
c)
F2 ( x )
2. a) Plot the points for F6 ( x) .
Sketch and label the graph F6 ( x) .
b) Similarly, sketch and label the graphs of F2 ( x) .
c) Use your table values and reasoning to
F6 ( x)
sketch and label the graph of F x .
F2 ( x )
F6 ( x)
3. Determine F x algebraically.
F26 2 , mT F2 2 , and mT F 2 35
6.6.1: Investigating the DivisionF2 ( x )
c)
F6 ( x ) sketch
F2 ( x )
and label the graph of F x .
F6 ( x)
F2 ( x )
3. Determine F x algebraically.
F6 2 1 , mT F6 1 , and mT F 1� 36
6.6.1: Investigating the Division of functions (continued)
Pair D F8 x x 2 , F5 x log3 x
1. Fill in the following table of values
1
x –1 0 1 2 3 9
2
a) F8 ( x)
b) F5 ( x)
F8 ( x )
c)
F5 ( x )
2. a) Plot the points for F8 ( x) .
Sketch and label the graph F8 ( x) .
b) Similarly, sketch and label the graph of F5 ( x) .
c) Use your table values and reasoning to
F8 ( x )
sketch and label the graph of F x .
F5 ( x )
F8 ( x )
3. Determine F x algebraically.
F58 37
Unit 6: Day 7: Compositions of Functions Numerically and Graphically MHF4U
Math Learning Goals Materials
Determine the composition of functions numerically and graphically. BLM 6.7.1, 6.7.2,
Connect transformations of functions with composition of functions. 6.7.3
chart paper
Explore the composition of a function with its inverse numerically and graphically,
graphing
and demonstrate that the result maps the input onto itself. technology
75 min
Assessment
Opportunities
Minds On… Whole Class Discussion
Model the use of a function machine presented on BLM 6.7.1 with an example.
Introduce the
Pairs Investigation notation
Assign each pair of students 2 values of ―x‖ from the given domain on y g f x to
BLM 6.7.1 which demonstrates: represent
1) function machines, composition of two
functions, the output
2) numerical and graphical representation of composition, of the table.
3) y f g x versus y g f x .
Students plot the results of their work on a large graph with f x and g x
already plotted (BLM 6.7.1).
Whole Class Discussion
Each student plots ordered pairs (Input(A), Output(B)) aka x, f g x from the Complete plotting
class graph on their individual graph on BLM 6.7.1. and discussion for
Lead a discussion that includes ideas such as: domain and range of all three y g f x before
functions, the relationship of the composition graph to originals and f g x completing
y f g x ,
versus y g f x .
(BLM 6.7.1, p. 2).
Action! Pairs Exploration
Pairs complete BLM 6.7.2 using graphing technology, as required.
Reasoning/Observation/Mental Note: Observe students' facility with the
inquiry process to determine their preparedness for the homework assignment.
Whole Class Discussion Students will do
Reinforce earlier findings and explore the question: further exploration
using algebra on
―When is f g x g f x ?‖ using the results of BLM 6.7.2. Day 8.
Consolidate Pairs Discussion
Discuss solutions
Debrief Students consolidate key concepts of Day 7 (BLM 6.7.3). on Day 8 (see
Model an example of a question requiring the answer of Always, Sometimes, or BLM 6.8.2).
Never (e.g., When you subtract you always get less than you started with).
Explore the following statement by analysing the class graphs and activating
prior knowledge re: transformations of parabolas. The composition, g f x , of a See Winplot file on
Day 9.
linear function of the form f x x B with a quadratic function, g x , will
This will be re-visted
always result in a horizontal translation of ―B‖ units. Ask: What if the on Day 9.
composition was f g x ? Extend the discussion to include linear functions of
the form f x A x B . How does the linear function predict the transformation
that occurs in the composition?
Curriculum Expectations/Anecdotal Feedback: Observe student readiness for
future discussion about the relationship between linear transformations and
composition.
Exploration Home Activity or Further Classroom Consolidation
Application Complete the Worksheet and be prepared to discuss your solutions.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 38
6.7.1: Great Composers!
f x x 2 g x x2 5
Partner A: You are f x
Determine the value of f x for a given value of x.
Give the value of f x to Partner B.
Partner B: You are g x
Partner A will give you a value.
Determine the value of g Input(A) Output(A)→Input(B) Output(B)
-3
x f x g f x
4 3 6
2
1 5
7
-1
f(x) = x - 2
2
g(x) = x - 5
? ?
? ?
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 39
6.7.1: Great Composers! (continued)
f x x 2 g x x2 5
Partner A: You are g x
Determine the value of g x for a given value of x.
Give the value of g x to Partner B.
Partner B: You are f x
Partner A will give you a value.
Determine the value of f
-3 Output(A)→Input(B) Output(B)
Input(A)
f g x
3
4
2
6 x g x
1 5
7
-1
2
g(x) = x - 5
f(x) = x - 2
?
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 40
6.7.1: Great Composers! (Teacher)
The domain of the composite function is x | 3 x 7 and the range of the composite
function is y | 5 y 20 . Provide a large grid and table of values that captures this
domain and range. Include the graphs of the original functions f x and g x for comparisons.
Graphs of f x , g x
f g x and g f x .
g f x
f x
g x
f g x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 41
6.7.2: Graphical Composure
f x 2x g x cos x
1. Using the model of the function machine below, complete the table of values for the
specified functions.
x g x f g x
x
2
3
g(x) = cos(x) 2
g(x) 2
0
f(x) = 2x
2
3
f(g(x)) 2
2
2. Sketch the graphs of the functions, y f x and y g x on the grid below.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 42
6.7.2: Graphical Composure (continued)
3. Using the table values and your sketch from Question 2, predict the graph
of y f g x 2cos x . Sketch your prediction on the previous grid.
4. Plot the values of y f g x from your table on the previous grid. Compare with your
prediction.
5. Use graphing technology to graph f g x . Sketch a copy of this graph on the grid below
and compare it to your predicted graph.
6. If your graph is different from the one created using technology, analyse the differences
and describe any aspects you did not initially think about when making your sketch. Explain
what you understand now that you did not consider.
7. If your graph is the same as the one created using technology, explain how you
determined the domain and range.
8. Use graphing technology to determine the validity of the following statement:
"The graph of y g f x f g x , when f x 2 x , and g x cos x .
Compare and discuss your answer with a partner.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 43
6.7.2: Graphical Composure (Teacher)
Note: The curves are not congruent. The rates of change differ.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 44
6.7.3: How Did We Get There?
With a partner, answer each of the following questions.
1. If the ordered pairs listed below correspond to the points on the curves g x and f g x
respectively, complete the second column of the chart for f x .
g x f x f g x
(0, -3) (0, 10)
(1, 5) (1, 2)
(2, 7) (2, 2)
(3, 9) (3, 10)
(4, 11) (4, 26)
(5, 13) (5, 50)
(6, 15) (6, 82)
2. Given two functions f x and g x such that g 2 7 and f g 2 50 .
Determine f 7 _______.
3. State if each of the following statements is:
always true (A), sometimes true (S), or never true (N).
Justify your answer using examples or reasoning. (Graphing technology is permitted)
a) The composition, g f x , of a linear function of the form
f x x a with an exponential, logarithmic, polynomial or sinusoidal A S N
function, g x , will result in a horizontal translation of "a" units.
b) For the composition y f g x , the range of f x is the domain of
A S N
g x .
c) f g x g f x A S N
d) If f g x g x then f x x A S N
e) The composition of two even functions will result in an even function. A S N
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 45
Unit 6: Day 8: Composition of Functions Algebraically MHF4U
Math Learning Goals Materials
Determine the composition of functions algebraically and state the domain and BLM 6.8.1, 6.8.2
graphing
range of the composition.
Connect numeric graphical and algebraic representations.
technology
Explore the composition of a function with its inverse algebraically.
75 min
Assessment
Opportunities
Minds On… Whole Class Discussion
Debrief solutions from Home Activity question 3 (BLM 6.7.3). Four Corners
Strategy – see
Identify three corners of the room to represent A (Always True), S (Sometimes p. 182, Think
True), and N (Never True). One at a time, students go to the corner that matches Literacy: Cross-
Curricular
their solution to that part of the Home Activity question and discuss in their Approaches
groups. A volunteer shares the groups' reasoning with students in other corners. Grades 7-12.
Pairs Exploration
Students explore function evaluation connecting to algebraic composition Students can switch
(BLM 6.8.1). positions after
listening to the
reasoning.
Action! Whole Class Instruction
Establish the procedure for composing two functions algebraically. Clarify
possible restrictions on the domain and range under composition. Use the
functions from Day 7 to demonstrate algebraic composition. Point out the
connections between the graphical representation and the algebraic
representation of composition.
Pairs Investigation Review finding the
Students complete BLM 6.8.2 using graphing technology as required. inverse of a function
algebraically and
Learning Skills (Teamwork)/Observation/Checklist: Observe and record graphically.
students' collaboration skills.
Whole Class Instruction
Complete the composition of the functions algebraically, noting the restrictions
on the domain of y log x (BLM 6.8.2).
Consolidate Whole Class Discussion
Debrief
Explore further examples and lead discussion to generalize the results f f 1 x
and f 1 f x
, namely, f f 1 x f 1 f x x (BLM 6.8.2). Examine possible
restrictions on the domain and range.
Home Activity or Further Classroom Consolidation
Exploration Complete additional procedural questions to determine f g x , g f x , f 1 x ,
Application
and f f 1 x .
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 46
6.8.1: Evaluating Functions
f x x 3 g x x 3
1. Evaluate the functions y f x and y g x by completing the table of values below.
x f x x 3 x g x x2
–2 –2
3 3
a –a
–b b+2
2 –4
x2 x+3
g x f x
2. Discuss with your partner the meaning of the notation y f g x . Summarize your
understanding below. Use examples, as necessary.
3. Compare the entries in the last two rows of the table for y f x if you were given specific
numerical values of x. Does your answer change for the last two rows of the table
for y g x ? Explain.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 47
6.8.2: Maintain Your Composure
f x log x g x 10 x
1. Use the model of the function machine below, complete the table of values for the specified
functions for values of x such that 5 x 5 .
x
x g x f g x
g(x) = 10x
g(x)
f(x) = log(x)
f(g(x))
2. What is the relationship between the domain and range of f g x ?
3. Use graphing technology to graph y f x , y g x , and y f g x .
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 48
6.8.2: Maintain Your Composure (continued)
4. Draw a line on the graph that would reflect the graph of f x onto g x .
What is the equation of this line?
5. How is the equation of the line you drew related to y f g x ?
6. Use your prior knowledge of these functions and the function machine model given in Step 1
to explain the relationship between the input value and the output value for y f g x .
7. Identify another pair of functions that have the same result as Step 6.
8. Is the following statement always true, sometimes true or never true?
Discuss your answer with a partner.
Given two functions, f x and g x such that g x f 1 x , then f g x x .
A S N
9. Use graphing technology to graph the composition y g f x identified at the beginning.
Compare this graph to the graph of y f g x in Step 3. Explain why the domain and
range of the graphs are different.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 49
6.8.2 Maintain Your Composure (Teacher)
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 50
Unit 6: Day 9: Solving Problems Involving Composition of Functions MHF4U
Math Learning Goals Materials
Connect transformations of functions with composition of functions. BLM 6.9.1
graphing
Solve problems involving composition of two functions including those from
real-world applications. technology
Reason about the nature of functions resulting the composition of functions as
even, odd, or neither.
75 min
Assessment
Opportunities
Minds On… Whole Class Discussion
Determine algebraically the composition of the functions, y x 2 and
g x x2 5
from Day 7. Discuss the graphical transformations of the parabola
which resulted from the composition of the quadratic function with the linear Note: The
function. relationship between
transformations and
Small Group Investigation composition involving
Students use graphing technology to explore the results of the composition a linear function was
of y f g x , where g x x B and f x is one of the following functions: explored on Day 7
with a quadratic
polynomial, exponential, logarithmic, or sinusoidal. Extend the exploration for function.
linear functions of the form g x A x B .
Use Winplot file for
Recall the statement and question posed on Day 7: The composition, g f x , of class demonstration.
a linear function of the form f x x B with a quadratic function, g x will U6L7_8_9.wp2
always result in a horizontal translation of ―B‖ units.
Ask: What if the composition was f g x ? Extend the discussion to include
linear functions of the form f x A x B . How does the linear function predict
the transformation that occurs in the composition? What is their position on this
statement now? Discuss.
Reasoning and Connecting/Observation/Checkbric: Listen to students'
reasoning as they investigate composed functions with respect to transformations
and make connections to the original functions.
Action! Individual Investigative Practice
Students complete BLM 6.9.1.
Circulate to clarify and guide student work.
Consolidate Whole Class Discussion
Debrief Consolidate the concepts developed on composition of functions. Share solutions
(BLM 6.9.1, particularly Question 6). Make conclusions about even/odd nature
of the composition as related to the even/odd nature of the original functions.
Reasoning and Connecting/Observation/Checkbric: Listen to students'
reasoning as they investigate composed functions with respect to even/odd
behaviour and make connections to the original functions.
Home Activity or Further Classroom Consolidation Provide additional
Application Graph the composition of two functions using graphing technology and solve questions and a
this problem involving a real-life application. problem.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 51
6.9.1: Solving Problems Involving Composition
1. If f g x log x 1 , determine expressions for f x and g x where g x x .
2
2. Given f x 2 x 3 , determine f 1 x algebraically. Show that f f 1 x x .
Explain this result numerically and graphically.
3. If f x 2 x 2 5 and g x 3 x 1 :
a) Determine algebraically f g x and g f x . Verify that f g x g f x .
b) Demonstrate numerically and graphically whether or not the functions resulting from the
composition are odd, even, or neither. Compare this feature to the original functions.
Verify your answer algebraically and graphically.
c) Describe the transformations of the parabola that occur as a result of the composition,
y f g x . Use a graphical or algebraic model to verify your findings.
d) Using graphing technology generalize your findings for part (c) for a linear function
y A x B .
4. Consider the functions, h x 2 x 2 7 x 5 and g x cos x :
a) Determine algebraically h g x and g h x .
b) Using graphing technology demonstrate numerically and graphically whether or not the
functions resulting from the composition are odd, even or neither.
Compare to the original functions.
5. The speed of a car, v kilometres per hour, at a time t hours is represented by
v t 40 3t t 2 . The rate of gasoline consumption of the car, c litres per kilometre,
2
v
at a speed of v kilometres per hour is represented by c v 0.1 0.15 .
500
Determine algebraically c v t , the rate of gasoline consumption as a function of time.
Determine, using technology, the time when the car is running most economically during a
four-hour trip.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 52
6.9.1: Solving Problems Involving Composition (continued)
6. Explain the meaning of the composition, y f g x for each of the following function pairs.
Give a possible "real-life" example for each.
g x f x
Velocity is a function of Time Consumption is a function of Velocity
Consumption is a function of Velocity Cost is a function of Consumption
Earnings is a function of Time Interest is a function of Earnings
Cost is a function of Consumption Interest is a function of Cost
Height is a function of Time Air Pressure is a function of Height
Depth is a function of Time Volume is a function of Depth
Sum of the Angles of a regular polygon Size of each Angle is a
is a function of the Number of Sides function of the Sum
Radius is a function of Time Volume is a function of Radius
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 53
6.9.1: Solving Problems Involving Composition (Teacher)
Solution to Question 4
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 54
Unit 6: Day 10: Putting It All Together (Part 1) MHF4U
Math Learning Goals Materials
Make connections between key features of graphs (e.g., odd/even or neither, zeros, BLM 6.10.1,
maximums/minimums, positive/negative, fraction less than 1 in size) that will have 6.10.2
graphing
an affect when combining two functions from different families.
technology
Identify the domain intervals necessary to describe the full behaviour of a
combined function.
Understand graphs of combined function by reasoning about the implication of the
key features of two functions, and make connections between transformations and
75 min composition.
Assessment
Opportunities
Minds On… Whole Class Discussion
Present the plan for Days 10, 11, and 12. Math congress
questions have some
Discuss the purpose of these days, details of each station, and the structure of the overlap so teachers
―math congress:‖ each group presents to another group one of the assigned can select groups to
present to each other
combination or composition of functions (BLM 6.10.1). accordingly.
Determine which question to present for assessment of the mathematical
processes. The ―receiving‖ group assesses knowledge and understanding. Some math congress
questions are
Questions are posed and answers are given between the groups. intentionally the
Share rubrics for teacher and peer assessment (BLM 6.11.6 and 6.11.7) or same as the function
develop with the class. combinations on the
card game on
Assign each group of four their three questions. Day 11.
Action! Groups Discussion and Planning
Groups B, E, and H
Groups work on their three assigned questions (BLM 6.10.1) making of the math congress
connections to their prior knowledge on even and odd functions and select each contain a
appropriate tools to justify their results graphically. composition of
functions that
Groups discuss, organize, and plan their presentation. assesses expectation
D2.8.
Connecting/Observation/Mental Note: Observe students facility to connect
prior learning on even and odd functions with combination of functions.
Consolidate Whole Class Discussion
Debrief Lead a discussion of even and odd functions as they relate to the combination or
composition of the functions.
Select two groups to present to one another's group (Station 4) at the congress at
the beginning of Day 11.
Assign two groups to each Station 1, 2, and 3 for the beginning of Day 11.
Home Activity or Further Classroom Consolidation
Exploration Complete and prepare to discuss with the group solutions to the congress
Application questions.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 55
6.10.1: Math Congress Questions
For each of your identified questions:
Fully analyse the combination of the functions algebraically and graphically by considering
appropriate domain, zeros, intercepts, increasing/decreasing behaviour, maximum/minimum
values, relative size (very large/very small) and reasoning about the implications of the
operations on the functions.
Identify whether the original functions are even, odd, or neither, and whether the combined
function is even, odd, or neither, algebraically and graphically.
Hypothesize a generalization for even, or odd, or neither functions and their combination.
Be prepared to discuss one of your question groups to a panel of peers for the math
congress.
Be prepared to respond to and ask questions of a panel of peers as they present one of
their questions to you.
Group A Group B
1. f x 2 x ; g x cos x; f x g x 1. f x 2 x ; g x cos 2 x ; g x cos x ; f x � x 2 ; g x log x ; f g x 3. f x log x ; g x 2 x 6; f g x
Group C Group D
1 x; 2x ; g x x 2;
g x
3. f x sin x ; g x 2 ; f g x
x
3. f x x 2 4; g x sin x ; f g x
Group E Group F
1. f x x 3 ; g x x; x 2 ; g x cos x ; sin x ; g x 2 x ;
g x
3. f x sin x ; g x 2 x 6; f g x 3. f x log x ; g x x 4; f g x
2
Group G Group H
1. f x sin x ; g x 2 x ; f x g x 1. x sin x ; g x 2 x ; g x f 2 x ; f x � 2x ; g x x 2; f g x 3. f x x ; g x 2 6; f g x
3 x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 56
6.10.2: Composition Solutions for Math Congress Questions
(Teacher)
The graphs of the composition of functions are for the math congress presentation.
Students may use technology to generate these graphs. They are assessed on the analysis of
the result.
Note: Graphs for the math congress questions for addition, subtraction, multiplication, and
division are on the BLM 6.11.5 (Teacher).
Example to use with class:
sin x and 2 x y2
sin x
The exponent of the composition is sin x , thus the value of the exponent is between –1
and 1, therefore the y-values of the composition function will oscillate between
1
2
i.e., 2 1 and
2 i.e., 21 over the same interval.
When the y-value of the composed function is 1, it corresponds to the x-intercepts of the
sine function, i.e., 2°.
The cyclic nature of the composition connects to the cyclic nature of the sin x , i.e., same
period.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 57
6.10.2: Composition Solutions for Math Congress Questions
(continued)
2x and x 2
2
y 2x
sin x and x 2 4 y sin2 x 4
log x and 2 x 6 y log 2 x 6
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 58
6.10.2: Composition Solutions for Math Congress Questions
(continued)
log x and x 2 4
y log x 2 4
log x and x 2 4 y log2 x 4
log x and x 2
y log x 2
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 59
6.10.2: Composition Solutions for Math Congress Questions
(continued)
sin x and 2 x
y sin 2x
sin x and 2x 6 y sin 2 x 6
y 2x 6
3
x3 and 2x 6
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 60
Unit 6: Days 11 and 12: Putting It All Together (Part 2) MHF4U
Math Learning Goals Materials
Consolidate applications of functions by modeling with more than one function. graphing
Consolidate procedural knowledge when combining functions. technology
BLM 6.11.1–
Communicate about functions algebraically, graphically and orally
6.11.8
Model real-life data by connecting to the various characteristics of functions.
CBR
Solve problems by modelling and reasoning. pendulum
interactive white
board, overhead,
75 min or chart paper
Assessment
Opportunities
Minds On… Small Group Organization
Students gather at Stations 1, 2, 3, or 4 as assigned on Day 10 – two groups per
station. (See BLM 6.11.1–6.11.5.) See BLM 6.11.6 –
Demonstrate an example of the card game at Station 3 for the whole class Game Answers.
(BLM 6.11.4).
Review the purpose of these days, the structure of the ―math congress,‖ and
details of each Station:
1. Data modeling with more than one function (Application; 30 minutes).
2. Procedural/practice questions (Knowledge; 30 minutes)
3. (i) Card game to identify pairs of cards that represent the combination of
functions graphically and algebraically. (Communicating, Representing,
and Reasoning; 15 minutes)
(ii) Prepare for ―congress.‖ Each group presents to another group one of the
assigned combination or composition of functions. (Communicating,
Representing, and Reasoning; 15 minutes)
4. Presentations through math congress (Knowledge and Processes;
30 minutes)
Action! Small Group Task Completion
Groups discuss, plan, and work at their assigned station for the allowed time.
Note: The first
During the congress, each group presents to the other (15 minutes each), and groups presenting
asks and answers questions. will not have
additional time during
Alert the class to rotate after 30 minutes. class to prepare their
presentation. The
Reasoning and Representation/Rubric/Anecdotal Notes: Listen to the teacher may
presentation of combined functions as students present to their peers rearrange times to
(BLM 6.11.7 and 6.11.8). allow for this.
Communicating and Reflecting/Observation/Anecdotal Notes: Listen to the
questions posed and reflections made on the presentation as students articulate
questions to the presenting group.
Consolidate Whole Class Discussion
Debrief Clarify questions, if needed.
Review the Day 12 plan to continue working in stations not visited or prepare for
course performance task and exam.
To prepare for the environmental context of the course performance task have
students brainstorm some words dealing with natural disasters. See Course
Performance Task Day 1 for samples.
Home Activity or Further Classroom Consolidation
Exploration
Application
Generate words dealing with the environment to post on the word wall.
Review for the course summative performance task and exam.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 61
6.11.1: Station Materials (Teacher)
Teacher arranges the room setup with materials for four Stations:
1. BLM 6.11.2, CBR and pendulum.
Data source for Q.2- E-STAT Table 053-0001, V62,
2. BLM 6.11.3 with additional questions created using course resources/texts.
3. BLM 6.11.4–6.11.6. Game cards are reproduced without the algebraic representation –
included for teacher reference only.
Functions in the game can be repurposed by comparing all combinations of graphs from a
given pair of graphs.
4. Tables, chairs and resources set up for congress presentations and questioning
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 62
6.11.2: Station 1: Application of Combination of Functions
1. Gathering Data: Pendulum Swing
Hypothesise the graph of the distance of the pendulum from CBR as it swings over a
time of 15 seconds.
Arrange the CBR and pendulum so the motion of the pendulum is captured in the CBR.
Record the motion for 15 seconds.
Sketch the graph of the motion of the pendulum and compare to your hypothesis.
Discuss the result and any misconceptions you may have had.
What two types of functions are likely represented by the motion of the pendulum?
Determine the combination of those functions to fit the graph as closely as possible.
2. Data and graphs "Baby Boom Data"
The data represents the quarterly number of births during the peak of the baby boom.
What two types of functions are likely represented here?
Determine the combination of those functions to fit the graph as closely as possible.
If the graph were to continue, when would the number of births fall below 10 000?
Reference –
Data source: E-STAT. Table 053-0001, vector v62,
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 63
6.11.2: Station 1 (continued)
3. Let F t represent the number of female college students in Canada in year t and
M t represent the number of male college students in Canada in year t. Let C t represent
the average number of hours per year a female spent communicating with peers
electronically. Let P t represent the average number of hours per year a male spent
communicating with peers electronically.
a) Create the function A t to represent the number of students in college in Canada in
year t.
b) Create the function G t to represent the number of hours all female college students
spent communicating with peers electronically in year t.
c) Create the function H t to represent the number of hours all male college students
spent communicating with peers electronically in year t.
d) Create a function T t to represent the total number of hours college students spent
communicating with peers electronically in year t.
4. The graph shows trends in iPods sales since 2002.
a) According to this data, when did the peak sales occur? Hypothesize why this was the
peak during this time period.
b) Over what 4-month period was the greatest rate of change?
c) If you were to describe this data algebraically, into what intervals would you divide the
graph and what function type would you choose for each interval? Justify your answers.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 64
6.11.3: Station 2: Procedural Knowledge and Understanding
1. Given f x x 2 2 x 1, g x 4 sin x , h x 3 x , determine the following and state any
restrictions on the domain:
a) f x g x
g x
b)
f x
c) f h x
1
2. Given f x
2
x 3, x 0, determine f 1 x . Show that f f 1 x x in more than one
way, using graphing technology.
3. Solve graphically and algebraically: x 3 2 x 7
2
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 65
6.11.4: Station 3: Card Game: Representations
Your team must find a pair of matching cards. To make a matching pair, you find one card that
has the graphs of two functions that correspond with a card that shows these functions
combined by an operation (addition, subtraction, multiplication, or division).
When you find a matching pair, state how the functions were combined. Discuss why you think it
is a match.
Check the answer (BLM 6.11.5) and reflect on the result, if you had an error.
Continue until all the cards are collected.
Some of the features to observe in finding a match are:
intercepts of combined and original graphs
intersections of original graphs
asymptotes
general motions, e.g., periodic, cubic, exponential
large and small values
odd and even functions
nature of the function between 0 and 1, 0 and –1
domain and range
Examples
The initial graph of sin x and 2 x , can be combined to produce the graphs shown below it.
Determine what operations are used to combine them and explain the reasoning. Check
answers after you have determined how the functions were combined.
sin x and 2 x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 66
6.11.4: Station 3: Card Game: Representations (continued)
Answers
sin x 2 x 2 x sin x
Periodic suggests sine or cosine Periodic suggests sine or cosine
Dramatic change for positive x-values, Dramatic change for positive x-values,
not existing for negative x-values, not existing for negative x-values,
suggests exponential suggests exponential
y-intercept of 1 can be obtained by x-intercepts correspond to the
adding the y-intercepts of each of the x-intercepts of the sine function therefore
original graphs, only addition will produce multiplication or division.
this Division by exponential would result in
small y-values in first and fourth
quadrant, division by sinusoidal would
result in asymptotes, therefore must be
multiplication.
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 67
6.11.5: Card Masters for Game
2 J
x and x3 x3 x
3 P
x
x and x 2 4
x2 4
4 L
2 x and cos x 2 x cos x
S G
cos x 2 x 2 x cos x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 68
6.11.5: Card Masters for Game (continued)
5 T
sin x and log x log x sin x
M H
log x sin x sin x log x
6 F
x and sin x x sin x
7 R
x 2 and sin x x 2 sin x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 69
6.11.5: Card Masters for Game (continued)
8 N
x 2 and cos x x 2 cos x
9 K
x2
2x and x 2
2x
Q 9
2x
2x and x 2
x2
4 4
2 x and cos x 2 x and cos x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 70
6.11.5: Card Masters for Game (continued)
5 5
sin x and log x sin x and log x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 71
6.11.6: Game Answers difference of odd functions is an odd function
cannot be multiplication since odd multiplied by
odd is even
general motion is cubic, result is cubic
0,0 is a point on both originals and
combination
cannot be division since no asymptote occurs
x-intercepts occur where the graphs intercept,
implying subtraction
2. x and x 3 J. x 3 x
asymptotes at 2 and –2 suggests division by
x2 4
division results in y-values of 1 on the combined
graph for values of x where the original graphs
intersect
when 0 y 1 the y-values of the combined
graph becomes large, and when 1 y 0 the
y-values of the combined graph becomes small
0,0 is a point on the combined graph giving
information about the numerator
odd function divided by an even function is an
odd function
x
P.
3. x and x 2 4
x 4
2
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 72
dramatic change for positive x-values, not
existing for negative x-values, suggests
exponential
y-intercept of 2 can be obtained by adding the
y-intercept of 1 of each of the original graphs,
only addition will produce this result
4. 2 x and cos x L. 2 x cos x decreases
quickly, suggesting subtraction of an exponential
4. 2 x and cos x S. cos x 2 x increases
quickly, suggesting subtraction of the periodic
from the exponential
4. 2 x and cos x G. 2 x cos x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 73 domain 0 suggests log function
decreasing sin x graph suggests something is
being "taken away," thus subtraction
x-intercepts are where the original graphs
intersect implying subtraction
when log is very small (or large negative), the
combined graph becomes very large, implying
subtraction the log values
5. sin x and log x H. sin x cos x
periodic suggests sine or cosine function
domain 0 suggests log function
x-intercepts are where the original graphs
intersect implying subtraction
when log is very small, the combined graph
remains small, implying subtraction from the log
5. sin x and log x M. log x sin x
periodic suggests sine or cosine function
domain 0 suggests log function
when log is very small, the combined graph
remains small, implying log is not being
subtracted
the sine curve is increasing, implying something
is being added to the sine.
5. sin x and log x T. log x sin x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 74 x-intercepts exist wherever there is an x-intercept
in either of the original functions, suggesting
multiplication
odd function multiplied by an odd function,
results in an even function
6. x and sin x F. x odd function,
results in an odd function
7. x 2 and sin x R. x 2 even function,
results in an even function
8. x 2 and cos x N. x 2 cos x
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 75 x-intercept occurs at x-intercept of x 2 suggesting
multiplication or division
where 2 x is small the combined graph is large
and vise versa, suggesting division by 2 x
where the graphs intersect at 2,4 , division
produces the point 2,1
x2
9. 2x and x2 K.
2x
asymptote at y-axis suggests division by a
function going through the origin
combined function is small as x gets small, and
is large as x gets large, suggest exponential
2x
9. 2x and x2 Q.
x2
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 76
6.11.7: Station 4 Rubric for Math Congress (Teacher)
Connecting
Criteria Below Level 1 Level 1 Level 2 Level 3 Level 4
Specific Feedback
Makes - makes weak - makes simple - makes - makes strong
connections connections connections appropriate connections
between between between connections between
information in the information in information in between information in
chart and the the chart and the chart and information in the chart and
graph. the graph the graph the chart and the graph
the graph
Gathers data - gathers data - gathers data - gathers data - gathers data
that can be used that is that is that is that is
to solve the connected to appropriate and appropriate and appropriate and
problem [e.g., the problem, connected to connected to connected to
select critical yet the problem, the problem, the problem,
x-values and inappropriate yet missing including most including all
intervals for the for the inquiry many significant significant
chart]. significant cases cases,
cases including
extreme cases
Reasoning and Proving
Interprets - misinterprets - misinterprets - correctly - correctly
graphs. a major part of part of the interprets the interprets the
the given given graphical given graphical given graphical
graphical information, but information, information,
information, but carries on to and makes and makes
carries on to make some reasonable subtle or
make some otherwise statements insightful
otherwise reasonable statements
reasonable statements
statements
Makes - makes - makes - makes - makes
inferences in the inferences that inferences that inferences that inferences that
chart about the have a limited have some have a direct have a direct
required graph. connection to connection to connection to connection to
the properties the properties the properties the properties
of the given of the given of the given of the given
graphs graphs graphs graphs, with
evidence of
reflection
Representing
Creates a graph - creates a - creates a - creates a - creates a
to represent the graph that graph that graph that graph that
data in the chart. represents little represents represents represents the
of the range of some of the most of the full range of
data range of data range of data data
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 77
6.11.8: Station 4 Rubric for Peer Assessment of
Math Congress (Student)
Communicating
Criteria Level 1 Level 2 Level 3 Level 4
Expresses and - expresses and - expresses and - expresses and - expresses and
organizes organizes organizes organizes organizes
mathematical mathematic mathematic mathematic mathematic
thinking with clarity thinking with limited thinking with some thinking with thinking with a high
and logical clarity (helpfulness) clarity (helpfulness) considerable clarity degree of clarity
organization using (helpfulness) (helpfulness)
oral and visual
forms.
Knowledge and Understanding
Interprets key - misinterprets a - misinterprets part - correctly - correctly
features of the major part of the of the given interprets the given interprets the given
graphs of the graphical aspects graphical graphical graphical
functions and of of the functions information, but information, and information, and
the combined and of the carries on to make makes reasonable makes subtle or
function. combined function some otherwise statements insightful
information, but reasonable statements
carries on to make statements
some otherwise
reasonable
statements
Makes inferences - makes inferences - makes inferences - makes inferences - makes inferences
in the chart about that have a limited that have some that have a direct that have a direct
the required graph. connection to the connection to the connection to the connection to the
properties of the properties of the properties of the properties of the
given graphs given graphs given graphs given graphs, with
evidence of
reflection
Representing
Creates a graph to - creates a graph - creates a graph - creates a graph - creates a graph
represent the data that represents that represents that represents that represents the
in the chart. little of the range of some of the range most of the range full range of data
data of data of data
TIPS4RM: MHF4U: Unit 6 – Modelling with More Than One Function 2008 78
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Teach algebra via programming?
In the 1980's, when Providence, Rhode Island tried College Board's Equity 2000, she served on the school board. "Business" and "consumer" math were eliminated in favor of algebra for all. The goal was to get everyone through geometry and advanced algebra. Providence assigned all sixth graders to pre-algebra.
The smart kids zipped through quickly, doing algebra in seventh, geometry in eighth and advanced algebra in ninth grade. Teachers created many levels of slower-paced classes for weaker students.
"In time, Equity 2000 got many more urban kids into college," but it only helped "kids for whom low expectations were the only real problem," Steiny writes. It will take "new approaches to lure students into the puzzles of mathematical reasoning."
My now-grown sons, two of whom became software developers, have been arguing since high school that learning computer software programming is essentially learning algebra, only infinitely more fun, interesting, and useful.
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Another stupid concept…in order to do most programming, a person (in this case the student) needs to understand the basic concepts of add, subtract, multiply, divide, modulus, and some basic logic in terms of how a program should flow, what to test for when input is supplied (#1 cause of all security holes in software, improper data validation), and a few other things.
Why do we as a nation want to try one fad after another when methods for teaching math like Singapore Math and Kumon work so well in countries that do far better in math than the U.S. of A. does?
Well, if you think of a computer program as a system for symbolically expressing the manipulation of data, then I guess programming and math come pretty close. Problems that students might want to solve (such as displaying 3D images in a game), require fairly serious linear algebra (4×4 transformation matrices) to understand, and might require physics, calculous and all sorts of other math. So, I suppose this would supply some motivation and the answer to the age-old question, "Why do we have to learn this?"
No it isn't. Both might require the ability to pay attention to detail, but skipping algebra closes career doors that are not opened by programming classes. As Steiny should know, even the New England Institute of Technology (a modern vocational-type school) requires algebra as a minimum for their technology degree programs.
"…only infinitely more fun, interesting, and useful."
Only for some people. How many people, who have difficulty with the details of algebra, are going to have the patience to create working and tested computer programs? Is this supposed to work because the material is more interesting? Once again, she is blaming the students; that all they need are motivation and engagement.
It would be better to argue with the CCSS pseudo-algebra II requirement for all. One should look at specific career path requirements. Many require more than algebra II and some will require less. However, it is wrong to suggest alternate paths for algebra without regard to future consequences.
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MTH212 Foundations of Elem Math II
Examines the conceptual basis of K-8 mathematics. Provides opportunities to experience using manipulatives to model operations with rational numbers including fractions, decimals, percents, and integers. Explores the set of irrational numbers, the set of real numbers, proportional reasoning,and simple probability and statistics. Includes content and mathematical practices based on the Common Core State Standards. Prerequisite: MTH 211 and its prerequisite requirements. Audit available.
(For detailed information, see the Course Content and Outcome Guide ).
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CHEGG TEXTBOOK SOLUTIONS FOR
Hutchinson s Basic Mathematical Skills With Geometry 8th Edition
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Although this text is designed to help you master the basic skills of arithmetic, it is occasionally preferable to perform complex calculations on a calculator. To that end, many of the exercise sets include a short explanation of how to use a calculator to do an operation described in the section. This explanation will be followed by a set of exercises for which the calculator might be the preferred tool. As indicated by the placement of the explanation, you should refrain from using a calculator on the exercises that precede it
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See quotes below; I have to jump in with what I know about this issue. (I have taught algebra to community college students for over 20 years; some of these students -- about 20% -- never had algebra previously.)
I think we are missing some key points. A variable is not simply a letter standing for a number. When we say "P = 2L + 2W", we are making a statement about a relationship between quantities, any one of which might be a "variable". Furthermore, not all "variables" imply that we will find a solution. Both of these issues are challenging to the first time student, but are critical if there is any hope of being able to apply algebra.
Further comments are given below the quotes.
numeracy@facteur.std.com wrote: > > Hi Bonnie, I would definitely start with concrete things. Remember the > blocks with hexagons and paarallelograms? You could use them or cuisenaire > rods or anything. After getting comfortable with the rrelative sizes > between them, assign a nmber value to a few. Then have her grab a handful > and count them up, i.e., four yellows, three orange ones, two greens. Then > write that out. 4Y +3O +2g. I've seen this work. Eventualllly she will > substitute with no fear. You will robabyl have to trick her into this by > tellling her it's parenting and not math so the anxiety doesn't get > triggered. Or tell her it's geometry, not algebra. > > Have fun!! Martha> Point: This transition is fairly easy, but is entirely too dangerous. A student says that "4Y" means "4 yellow ones", and makes two assumptions: 1) That the variable is always 1. 2) That the variable is a label. Hopefully, you can see the danger in the first assumption. For the second, look at the situation of working with money invested in accounts earning interest (say, at 5%). The "label" interpretation is that x is 5%; when I tell the student that the interest is ".05x", the student tells me this can't be true! If you use this type of transition to variables, you need to provide a corrective package later to make sure the student has a more complete understanding of "variable."
> >Has anyone run in to a case like the student I have had who seems unable > >to do any math that has unknowns or variables in it? She is mid 40s, > >very bright, English major going on to a Masters program. She can do all > >sorts of computations including fractions, percentages, ratios, and word > >problems are some of her favorite things to do. But as soon as you give > >her something like 4 + 2x - 6 + 5x= 95, she is totally frozen. She can't > >get past go when trying to combine like terms, and reacts physically > >(anxiety, tears, etc.) She has recently been tested (instrument unknown > >to me at this point) by our local Special Ed teacher to see if there is > >some way to pinpoint what the cognitive problem is. She has started math > >class with me three times, but hasn't been able to stay with it long > >enough for me to get to try some alternative approaches. > >I look forward to any thoughts you might have on this. > > > >Bonnie Fortini (14-e) I have seen this, as well. I would suggest, however, that the problem is not likely to be "cognitive" in the sense that there are skills or processes missing (or erroneous). The problem often deals with emotional trauma which has been paired up with "algebra" -- which is like a person who can't deal with a certain model of car because of an earlier trauma; it's not a property of algebra, but it is a strong association that is challenging to breakdown. Often, a student with moderate anxiety can be helped just by learning that most people feel math anxiety -- including their teachers. Situations involving more extensive trauma, however, are best passed along to counseling professionals.
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Hi , This morning I started solving my mathematics homework on the topic Intermediate algebra. I am currently unable to complete the same since I am unfamiliar with the fundamentals of radical expressions, exponent rules and solving inequalities. Would it be possible for anyone to assist me with this?
I could help you if you can be more specific and provide details about radar math formulas. A right software would be ideal rather than a costly algebra tutor. After trying a number of program I found the Algebrator to be the best I have so far found . It solves any algebra problem from your book . It also explains all the steps (of the solution). You can just reproduce as your homework assignment. However, you should use it to learn algebra, and simply not use it to copy answers.
Even I've been through times when I was trying to figure out a solution to certain type of questions pertaining to trinomials and x-intercept. But then I found this piece of software and it was almost like I found a magic wand. In the blink of an eye it would solve even the most difficult questions for you. And the fact that it gives a detailed and elaborate explanation makes it even more useful . It's a must buy for every algebra student.
Please do not take me wrong. I am searching for no shortcut. It's just that I don't seem to get enough time to try solving problems again and again. I will use the software as a guide only. Where can I get it?
Algebrator is a incredible product and is definitely worth a try. You will also find several exciting stuff there. I use it as reference software for my math problems and can swear that it has made learning math much more fun .
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Algebra: Form and Function
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Read More Algebraic equations describe the laws of science, the principles of engineering, and the rules of business. The power of algebra lies in the efficient symbolic representation of complex ideas, which also presents the main difficulty in learning it. It is easy to forget the underlying structure of algebra and rely instead on a surface knowledge of algebraic manipulations. Most students rely on surface knowledge of algebraic manipulations without understanding the underlying structure of algebra that allows them to see patterns and apply it to multiple situations: McCallum focuses on the structure from the start.
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Fair. Paperback. All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing. SKU: 9781118449196449196
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ib tutor The IB Diploma Programme mathematics higher level course is for students with a strong background in Mathematics and competence in a range of analytical and technical skills math online Students will be likely to include mathematics as a major component of university studies—either in its own right or within courses such as physics, engineering or technology. The course focuses on developing important mathematical concepts in a comprehensive, coherent and rigorous way through a balanced approach ib math Students are encouraged to apply their mathematical knowledge to solve problems set in a variety of meaningful contexts and to justify and prove results .
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Deep Dive into Mathematica's Numerics: Applications and Tips
Andrew Moylan
In this course from the Wolfram Mathematica Virtual Conference 2011, you'll learn how to best use Mathematica's numerics functions in advanced settings. Topics include techniques and best practices for using multiple numerics functions together, advanced numeric features, and understanding precision and accuracy
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