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Algebra, 1st Edition
ISBN10: 0-495-38798-3
ISBN13: 978-0-495-38798-5
AUTHORS: Kaufmann/Schwitters
INTERMEDIATE ALGEBRA'S simple, three-step problem-solving approach—learn a skill, use the skill to solve equations, and then use the equation to solve application problems—keeps you focused on building skills and reinforcing them through practice. This straightforward approach, in an easy-to-read format, has helped many students grasp and apply fundamental problem-solving skills. The carefully structured pedagogy includes learning objectives, detailed examples to help you see how concepts are used and applied, practice exercises, and helpful end-of-chapter reviews. Problems and examples reference a broad array of topics as well as career areas such as electronics, mechanics, and health, showing you that mathematics is part of everyday life
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Description
Calculus A introduces limits, differentiation, and applications of differentiation. The student will find and evaluate finite and infinite limits graphically, numerically, and analytically. The student will find derivatives using a variety of methods including the chain rule and implicit differentiation. Then the student will use the first derivative test and the second derivative test to analyze and sketch functions. Finally, the student will find derivatives using a variety of methods including substitution.
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Basic Remedial Mathematics Volume 1 text starts with Addition of positive real numbers and goes to decimals. This book is not intended to replace a beginning text book. Mathematics remediation is the most important consideration. However, it is not assumed that students have any prerequisite Skills. The first or second grade pupil must learn arithmetic from A very different perspective. However, in terms of remediation, Most students in K 9-12 may need a basic refresher course, but Not a new beginning. Students must read, think and apply basic mathematics concepts To practical situations. In the text, each concept is assessed on an individual basis. After the student has demonstrated proficiency on the individual Concepts, mastery is determined from the exit tests where a Variety of concepts must be solved.
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Course Frequency: Full-year course, five times per
week Credits Offered: Five Prerequisites: C- or higher in Elementary Algebra 1-1 (SP)
Background to the Curriculum This course uses the Glencoe Algebra 1 text, 1996 edition, and has been
updated to the 2000 edition. The Glencoe text replaced the Holt Algebra 1 and had been used for the previous 10 years. The Glencoe text is
followed quite closely since it matches both the 2000
edition of the National Council of Teachers of Mathematics Curriculum Standards
and the 2000 edition of the Massachusetts State
Framework recommendations for a first-year algebra course . This course exposes
students to the first year Algebra 1 curriculum and
is well aligned with national and state guidelines. Teachers bring in other
material where appropriate and make minor changes as to
the specific sections taught each year, after consultation with the RDL.
Assessment Students are generally assessed by in-class tests and quizzes, which are
administered regularly throughout a marking period.
Generally, two quizzes are equivalent to a test. The students' attitude, effort,
and quality of homework preparations will also impact
their term grade to a small degree. Teachers informally assess students every
day by asking pivotal questions, as well as questions
involving mechanics or concepts, and the students' term grades may be positively
affected to a small degree based on their responses.
A standardized midyear examination and final examination
are administered to all students in this course in order to assess their
longterm
retention of the course material.
Technology Learning Objectives Addressed in This Course
(This section is for faculty and administrative reference; students and parents
may disregard.)
Materials and Resources Teachers use other resources for supplementary ideas, such as the "Algebra
with Pizzazz" series of activities. There are review
materials that closely match most tests and quizzes, as well as a close
resemblance to the departmental examinations. All teachers of
the course use these materials. Teachers may also reinforce ideas by using
manipulatives, such as algebra tiles , in class.
MATH 225 CALCULUS II
Course Description:
This course is a continuation of Calculus I. Topics include an introduction to
differential
equations, techniques and applications of integration, L'Hopital's Rule,
improper
integrals, infinite series, conics, and parametric and polar equations.
Prerequisite: MAT 220 or its equivalent. 4 credits
Course Objectives
Upon successful completion of the course, the student will be able to define,
use
correctly, and apply to solving problems the concepts and terminology in each of
the
topics listed below. Upon successful completion of the course, students will:
1. Find general and particular solutions of differential
equations .
2. Use separation of variables to solve a simple differential equation.
3. Calculate Areas and Volumes by Integration.
4. Calculate arc length of a smooth curve.
5. Calculate the area of a surface of revolution.
6. Apply integration to various situations such as work, centers of gravity,
motion, etc.
7. Use techniques of integration such as integration by parts, trigonometric
substitution,
partial fractions.
8. Apply L'Hopital's rule to indeterminate forms.
9. Evaluate improper integrals.
10. Use an appropriate test to determine the convergence or divergence of
infinite
sequences and series.
11. Approximate a function with a Taylor or MacLaurin polynomial.
12. Find a power series representation of a function.
13. Find the equation of a parabola , ellipse, and hyperbola.
14. Sketch the graph of a curve defined by a set of parametric equations.
15. Understand the polar coordinate system.
Course Requirements
Students are expected to attend all scheduled classes, do the homework assigned
each day
for the next class, take tests, and be active participants in the class.
Non-discrimination and Disability Statements:
Southern Maine Community College is an equal opportunity /affirmative action
institution and
employer. For more information, please call 207-741-5798.
If you have a disabling condition and wish to request
accommodations in order to have
reasonable access to the programs and services offered by SMCC, you must
register with the
disability services coordinator, Mark Krogman, who can be reached at 741-5629.
(TTD 207-741-5667) Further information about services for students with
disabilities and the
accommodation process is available upon request at this number.
Course Evaluation: Students may evaluate the course
online and anonymously by going to
"Resources for Current Students" at the SMCC homepage and selecting "Evaluate
Your
Courses." The online course evaluation is available to students two weeks prior
to the end date
of the course. Students cannot see a course grade online until the online course
evaluation is
completed.
Learning Outcomes for Introductory Algebra
Course Description:
For students with limited algebra experience . Curriculum will include solving
algebraic
equations, operations with polynomials, and extensive factoring of polynomials .
Graphical
solutions for linear and quadratic equations will be explored , including slope,
intercepts, and
inequalities. Culminating material will include solving rational, radical, and
systems of
equations
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Solutions Index
These are solutions to the homework assignments. There may be
errors in them. Use them as a resource, not as the absolute
answer. If you find errors or possible errors, please email me an
let me know about them. My email address is kmiller@byu.edu.
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I could see how what I was learning would help me in future engineering classes. The material can be hard, and there is quite a lot to learn, but all it takes is consistent practice. I thought the material was important
The class more tricky than it was hard. Tests counted for most of the grade so you could easily mess up your average if you weren't prepared on a test day. Homework wasn't required, although I don't think most people could get through 242 without doing it. The attendance policy was fair and the class itself was fine as far as math classes go, but nothing to be excited about.
This guy was pretty good. He had really nice handwriting and he was very easy to understand. He knew the material very well and he wanted to help everyone understand the math, but I felt like the classroom environment was too "distant" even for a class that tiny. Also, the class averages were usually not very great, but that was more a factor of people not practicing enough rather than his instruction. He didn't give us a syllabus until a week into the class, too.
Nice guy. Hard to understand what he's doing sometimes. He tends to work through a problem without explaining how he's working through the problem. Tests are straightforward, but hard. Curves every test and counts the lowest one as half.
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Discrete Mathematics Study Materials
Discrete Mathematics
Lectures Ppt
Click on the blue colored links to download the lectures.
Course Description
This course covered the mathematical topics most directly related to computer science. Topics included: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. The analysis of algorithms requires the ability to count the number of operations in an algorithm. Recursive algorithms in particular depend on the solution to a recurrence equation, and a proof of correctness by mathematical induction. The design of a digital circuit requires the knowledge of Boolean algebra. Software engineering uses sets, graphs, trees and other data structures. Number theory is at the heart of secure messaging systems and cryptography. Logic is used in AI research in theorem proving and in database query systems. Proofs by induction and the more general notions of mathematical proof are ubiquitous in theory of computation, compiler design and formal grammars. Probabilistic notions crop up in architectural trade-offs in hardware design.
Lecture 1:What kinds of problems are solved in discrete math?What are proofs? Examples of proofs by contradiction, and proofs by induction:Triangle numbers, irrational numbers, and prime numbers. (3.1-3.2)
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MATHEMATICS PROGRAM MODELS FOR OHIO HIGH SCHOOLS
The ODE Mathematics Program Models offer six (6) different sequences of courses that take an applications,
blended, or connected approach to the high school mathematics curriculum. The ORC Pacing Guides (upper left navigation
bar) feature a schedule of topics, links to best practice lessons, teaching tips, and rich problems to engage students
in exploration, analysis, and application of big ideas in mathematics.
(Below is condensed from Ohio Department of Education draft, June 2006)
The State Board of Education adopted the Ohio Academic Content Standards for K-12 Mathematics in December 2001.
The Standards set high learning expectations for every student, recognizing that in the 21st century, every student
will need a strong preparation in mathematics. In Ohio, the assumption is that all students can learn significant
mathematics, and the commitment is that all students will be successful in learning mathematics and will graduate
from high school fully prepared for the demands of the workplace and further study.
Many factors influence how secondary mathematics programs can best be designed and delivered at this time.
Day-to-day decision making, as well as the expectations for today's workforce, require a greater emphasis on data
analysis, probability, and statistics in the secondary curriculum. The tools of technology make some mathematical
concepts accessible to students at an earlier stage. The curriculum of the middle grades now includes many of the
basic concepts of algebra, geometry, measurement, and data analysis. Consequently, what is needed in many Ohio
districts is not a simple adjustment on the margin of an old curriculum, but rather a full rethinking of the
secondary school mathematics program.
There are many ways a curriculum can be configured to respond to the requirements of the Content Standards.
In the area of secondary mathematics, the Department of Education is providing districts with three different
models for mathematics programs in grades 9-12.
Descriptions of the Mathematics Program Models
The three models were drafted by a panel of Ohio teachers, mathematicians, and mathematics educators in the
summer of 2005. They were reviewed and discussed by professional groups, practitioners, and others during the
school year 2005-2006, and after revision, are now available to schools. The models are presented in terms of
years of study (Year 1 through Year 5) rather than in terms of grade levels (grade 9 through grade 12), recognizing
that some students will start the secondary mathematics curriculum in grade 8, others in grade 9, and that there
can be years when some students take more than one mathematics course. The models emphasize the importance of
every student taking mathematics in each of the four years of high school, and they provide appropriate courses
for all students in grade 12.
Characteristics Common to All Three Program Models
Although the models presented here offer distinctive ways of approaching the mathematics described in the
Ohio Academic Content Standards, they share several basic characteristics:
Each demonstrates how the Standards can be implemented through a curriculum and how instruction can be
organized to improve student learning;
Each prepares students to achieve or exceed the proficiency level on the mathematics portion of the Ohio
Graduation Test in grade 10 and to achieve or exceed the requirements to enter Ohio college and university
mathematics
and logical reasoning;
Each displays the connectedness and coherence of the mathematics studied within each course and across
the courses in a sequence.
Distinctive Characteristics of the Three Models
Each model also has distinctive characteristics:
Model A. This model uses the applications
of mathematics to motivate
the need to master mathematical topics in algebra and geometry. By using applications to motivate the mathematics,
students can become more engaged in algebraic and geometric topics, and motivated to work hard on meaningful problems.
Mathematics developed in this way is intended to encourage problem solving and reasoning skills, thus preparing
students well for the workplace or for further education.
Model B. This model blends
the mathematics of the various content
strands (algebra/number, geometry/measurement, data/statistics), weaving them together in each course and providing
a sequence of courses that build on one another to form a coherent curriculum. Data topics are woven throughout
the model with a focus on a data project in Year 3.
Model C. This model features a
classic sequence of courses that emphasizes connections
across content strands. Data analysis topics have been added to the familiar high school mathematics curriculum. Year 1 focuses on algebraic thinking and skills, augmented with
data analysis. Year 2 focuses on geometric topics, both synthetic and analytic, and includes formal geometric
argument. Year 3 extends the algebra topics from Year 1 and introduces traditional topics of Algebra II. Year 4
includes trigonometric functions and other topics from pre-calculus mathematics.
Each of the Models A, B, and C prepares students to take a calculus course in their first year of college. The
Program Models presume that all Ohio graduates will enter postsecondary education at some time, but that not every
student's academic program will include calculus. Consequently, Models A, B, and C are followed by Model A', Model B', and
Model C', respectively, which adjust the original models to provide
an appropriate curriculum for students whose postsecondary program will not include the study of calculus.
Success for All Students
A program model is a guide to assist in organizing mathematical ideas and student experiences for effective learning.
However, different students learn in different ways. The amount of time, practice, and assistance students require to
learn mathematics varies from student to student. These differences must be accommodated in a district's plan for
delivering the curriculum. In this section, we offer suggestions for organizing programs to accommodate student
differences. We offer suggestions for three specific groups of students:
Students entering grade 9 without the mathematical skills and understanding needed to be successful in a Year
1 course;
Students who have completed grade 10 but not achieved or exceeded the proficiency level on the mathematics portion
of the Ohio Graduation Test;
Students with the background and abilities to be accelerated in the regular mathematics curriculum.
Preparation for the Year 1 Mathematics Course
A mathematics curriculum that reflects the Ohio Content Standards will build mathematical skills and dispositions that
enable all students to understand the fundamentals of algebra. As early as pre-kindergarten, algebraic thinking activities
such as finding patterns, identifying missing pieces in sequences, and acquiring informal number sense will be central parts
of students' experiences. The middle school curriculum moves students from numerical arithmetic to generalized arithmetic
where symbols can represent numbers. This curriculum gives students experience with numeric, geometric, and algebraic
representations of relationships. Students develop proportional reasoning skills; they investigate more complex problem
settings and move from concrete experiences in mathematics to the formulation of more abstract concepts.
The Year 1 mathematics course in any secondary curriculum model is expected to be the foundation for future learning of
mathematics. Formal algebra will be a focus of this course. Whether students enter the workforce directly after graduation
or enter postsecondary education, success in Year 1 mathematics will be critical to their futures. There are several
strategies districts should consider for students who complete grade 8 without the mathematics background needed to succeed
in Year 1. These strategies assure that all students study Year 1 mathematics no later than grade 9.
Summer Sessions
During the summer prior to their Year 1 course, students could attend:
A focused summer course that strengthens pre-algebra methods and terminology, provides a review of basic
mathematical procedures, and uses some topics of discrete mathematics to help students move from concrete thinking
to generalization, or
A computer-based program with a teacher or coach to individualize students' instruction and correct
misunderstandings.
During the Standard School Year
In addition to summer opportunities, districts may consider the following options:
Provide some Year 1 mathematics classes in grade 9 that meet 8 or 10 periods a week for students who need
more time to learn the mathematics in this course. Alternatively, all Year 1 mathematics classes can be taught
for 8 or 10 periods a week so teachers have time to differentiate instruction and engage in extended, supervised
problem solving.
Create a program of peer tutoring that includes training, supervision, and time for students to work with other
students.
Create Mathematics Labs associated with specific mathematics courses (similar to labs that are linked to science
courses) and to which students are assigned on a regular basis.
Create parent/community help teams that work under the direction of teachers and assist students with mathematics
after school or during study halls.
A common feature of these strategies is that each one recognizes some students will need more time and more assistance
to be successful in learning the mathematics of the Year 1 course. There are, of course, costs to each of these
interventions. However, the costs of providing timely help to students is significantly less than the cost of teaching
remedial courses or allowing students to enter the workforce with deficiencies in mathematics.
Reaching Proficiency Level on the OGT
Students who do not achieve or exceed the proficiency level on the mathematics portion of the Ohio Graduation Test in
grade 10 can benefit from the following options:
Require students to attend a summer program between grades 10 and 11 in which basic concepts are reviewed and problem
solving is emphasized. These students should re-take the OGT when it is offered again later in the summer.
Offer before school, after school, or Saturday sessions to review core mathematics topics and work with students
individually; study hall periods may be used in this way for some students.
Develop peer-tutoring programs to help students who did not succeed on the OGT, giving peer tutors sufficient training
and supervision.
Develop a 9-week OGT preparation course to be taken concurrently with the Year 3 mathematics course during the first
grading period in grade 11. This course could also be taught during the second semester in preparation for the spring
administration of the OGT. (Because the content of this short course will repeat content from earlier courses, credit
for this course should not count toward the mathematics credits required for graduation.)
Students Who Are Accelerated in the Curriculum
Some students are able to move successfully through a standard mathematics curriculum at a quicker pace than the majority
of students. Commitment to accelerated students must be as great as the commitment to other students to assure that they are
challenged in each year of study and persist in mathematics through their senior year. Two strategies are suggested:
A district may designate some sections of a regular course as honors or enriched and in these sections deal with
topics in greater depth, assign students more complex problems, and develop more team projects for students.
Differentiating instruction in this way, rather than having a student skip a course in order to move ahead,
will assure students do not miss critical material covered in each of the grade level curricula.
Some students may have the ability to study the Year 1 course in 8th grade if the curriculum has been modified
to assure they have studied all topics of the middle school curriculum before grade 8. Because the Ohio Academic
Content Standards in Mathematics identify new topics to be introduced in each of the middle grades, no mathematics
course can simply be skipped. Students with the potential to be accelerated will need to be identified by the
teaching staff and by readiness tests, and have their curriculum appropriately modified in the grades prior to
grade 8. Students who study the Year 1 course in 8th grade should move ahead to the Year 2 course in 9th grade,
continue in an enriched curriculum through grade 11, and study an advanced level mathematics course in grade 12
so they are well positioned for further study or for workplace opportunities.
Advanced Courses for Accelerated Students
The models present several options for accelerated students after they have completed the mathematics in the standard
curriculum. The models include a course called Modeling and Quantitative Reasoning that provides mathematics accessible
and of interest to high school students, but not always included in the high school curriculum. Another option for students
who have strong backgrounds in algebra, geometry, coordinate geometry, trigonometry, and pre-calculus mathematics is a
course in calculus. When a calculus course is offered for high school students, the course should be taught at the college
level and students should expect it to replace a first-year calculus course in college. This can be assured by using a
College Board Advanced Placement calculus course and requiring students to take the AP exam at the end of the course. In
some locations, accelerated students are able to enroll in a mathematics course at an area college or take a college level
course through distance education, concurrent with their high school studies. The models also prepare accelerated students
to take an Advanced Placement statistics course, which can be an exciting and appropriate option.
Mathematical Processes
The content in the mathematics Program Models is specified in the Ohio Academic Content Standards: Number, Number Sense
and Operations; Measurement; Geometry and Spatial Sense; Patterns, Functions and Algebra; Data Analysis and Probability.
The sixth standard, Mathematical Processes, is the thread that ties the five content standards together to make a meaningful
and cohesive curriculum. Mathematical processes can be divided into five strands: problem solving, reasoning, communication,
representation, and connections.
Authentic problem solving requires students not simply to get an answer but to develop strategies to
analyze and investigate problem contexts. The National Council of Teachers of Mathematics publication, Principles and
Standards for School Mathematics, states that "solving problems is not only a goal of learning mathematics but also
a major means of doing so. Students should have frequent opportunities to formulate, grapple with and solve complex
problems that require a significant amount of effort and should then be encouraged to reflect on their thinking."
Indeed, this is how students come to understand deeply the mathematical topics in their courses.
"Reasoning involves examining patterns, making conjectures about generalizations, and evaluating those
conjectures" (Ohio Academic Content Standards, K-12 Mathematics, p. 196). In mathematics, reasoning includes
creating arguments using inductive and deductive techniques. Students need opportunities to make and test their
conjectures, explain their reasoning, and evaluate the arguments of other students as well as their own.
Oral and written communication skills give students tools for sharing ideas and clarifying their
understanding of mathematical ideas. Mathematics has its own language, and this language becomes increasingly more
precise as students move through their studies. Developing skill in using this language requires students to read,
write, and talk about mathematics. Understanding mathematical terminology is essential to understanding mathematical
concepts.
Mathematics uses many different forms of representation to embody mathematical concepts and
relationships. Some are numerical (e.g., tables); some are algebraic (e.g., expressions, equations); some are geometric
(e.g., sketches, graphs); some are physical models. Students need to be comfortable using multiple representations for a
single concept. This skill will help them develop problem-solving strategies and communicate mathematical ideas
effectively to others. Appropriate use of technology is an essential tool for increasing students' access to different
kinds of representation in mathematics.
A coherent curriculum will help students make connections between the mathematical concepts they
learned in earlier grades and the concepts they study later on. Students need to appreciate that the five content
strands are not independent blocks of mathematics and that the process standard is part of learning within each content
strand. Without this appreciation, students may view the content of their courses as little more than a checklist of
topics. Students also need to experience the connections between mathematics and the other subjects they study. Their
mathematics courses should include frequent applications drawn from the life sciences, physical sciences, social studies,
and other fields. If students are to understand the importance and power of mathematics, these connections need to be
explicitly discussed.
In the Program Models, these mathematical processes are developed through course design and through experiences students
have when they work with rich contextual problems. Successful learning of mathematics requires that students struggle
with complex problems, communicate mathematics clearly, represent mathematics accurately and in various forms, make
conjectures and reason effectively, and connect mathematical concepts across the various areas of mathematics and to
applications in other fields. There is no shortcut. Each of the processes must be developed in every course, in every
sequence, and in every year of study.
Technology Assumptions
Appropriate use of technology in the mathematics classroom is an issue that must be addressed in the development of a
new curriculum. In this area, there are dual goals: (1) student proficiency with foundational skills and basic mathematical
concepts using basic manual algorithms, and (2) student competence in using appropriate technology to encourage mathematical
exploration and enhance understanding.
With respect to the first goal, the Program Models presume that students will enter the Year 1 course with an
understanding of basic mathematical concepts and with proficiency in performing accurate pencil and paper numerical
procedures. Even so, the secondary program should be designed to strengthen numerical skills and build additional skills
in algebraic computation, estimation, and mental mathematics. The study of algebra, measurement, geometry, and data
analysis provides useful contexts for students to continue to develop written and mental computational skills that deepen
their understanding of mathematics and strengthen their abilities in problem solving.
With respect to the second goal, the Program Models presume that students will use technology as a tool in learning the
mathematical concepts and working the complex problems in the secondary school curriculum. For example, technology can
assist students in investigating applications of mathematics, testing mathematical conjectures, visualizing transformations
of geometric shapes, and handling large data sets. Technology appropriately used can enhance students' understanding and
use of numbers and operations, as well as facilitate the learning of new concepts. Students will need to be alerted to
the possibility of serious round-off error when technology is used for complex computations in real-world applications.
At this time, the Ohio Graduation Test allows students to use a state-specified scientific calculator. This calculator
is primarily a computational tool, and students will need adequate time and practice using it prior to the OGT. A
scientific calculator alone does not provide all the features needed to study the topics described in the Program Models.
Implementing the Program Models requires decisions about the kinds of technology that students will use at different stages
of their learning to assure a balanced program that results in students' knowing when to use technology and when not to,
when to use pencil and paper, and when to do mathematics in their heads. The goal, always, is to develop a program that
focuses on mathematical understanding and proficiency.
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Mathematics for Elementary Teachers
9780470105832
ISBN:
0470105836
Edition: 8 Pub Date: 2008 Publisher: Wiley, John & Sons, Incorporated
Summary: Now in its eighth edition, this book masterfully integrates skills, concepts, and activities to motivate learning. It emphasizes the relevance of mathematics to help readers learn the importance of the information being covered [more
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Search Course Communities:
Course Communities
Lesson 7: Linear Systems
Course Topic(s):
Developmental Math | Systems of Equations
A calculator based introduction to systems of linear equations. Systems are solved using the graphing method: first by estimating the apparent intersection of the two lines and then later by using the intersect function on the calculator to find the exact solution. Inconsistent and consistent solutions are also discussed. There are both applications based problems and non applications based problems.
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Consumer Mathematics A, the first semester of a two semester series, focuses on basic math skills used in everyday life with the goal of developing intelligent consumers. The practical applications of math are studied using real world situations. Personal finances are emphasized through the study of personal earnings, the elements of business, credit, and life insurance. Prerequisites include Algebra I and Geometry.
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Cliffs Quick Review For Geometry - 01 edition
Summary: When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core geometry concepts -- from perimeter, area, and similarity to parallel lines, geometric solids, and coordinate geometry -- and get the best possible grade.
At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented wri...show moreters who know how to cut to the chase -- and zero in on the essential information you need to succeed. ...show less
Ed Kohn, MS is an outstanding educator and author with over 33 years experience teaching mathematics. Currently, he is the testing coordinator and math department chairman at Sherman Oaks Center for Enriched Studies.
2001Shopbookaholic Wichita, KSSusies Books Garner, NC
2001 Paperback This Book is in very good
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Find a Concordville can guide, but students must do the work.Introduction to the basics of symbolic representation and manipulation of variables. Liberal use of concrete examples. May require review of arithmetic concepts, including fractions and decimals
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Extended Mathematics for Cambridge IGCSE
'Extended Mathematics for Cambridge IGCSE' provides the first of a two-year course leading to the Cambridge IGCSE Mathematics Extended Level examination from University of Cambridge International Examinations. This is the second of two books (Core and Extended), which together completely cover the syllabus for the the Cambridge IGCSE Mathematics Extended Level.
Author: Simpson, A.
ISBN: 9780521186032
Published in 2011.
Edition: 1
Published by Cambridge University Press, India [New window] More information on Extended Mathematics for Cambridge IGCSE [New window]
Extended Mathematics for Cambridge IGCSE
Reinforcing all the basic principles and written with international students in mind, 'Extended Mathematics for Cambridge IGCSE' provides a complete course matching the extended level content of the Cambridge IGCSE 0580 Mathematics syllabus.
Author: Haighton, J, Manning, A, McManus, G, Thornton, M and White, K
ISBN: 9781408516522
Published in 2012.
Published by Nelson Thornes, UK More information on Extended Mathematics for Cambridge IGCSE [New window]
Extended Mathematics for Cambridge IGCSE - Teacher's Resource Kit (with CD)
This new Teacher's Resource Kit offers expert support for your Cambridge IGCSE teaching. The Teacher's Guide includes lesson plans and worksheets, while the Teacher's CD offers a host of customisable worksheets and ready-made editable PowerPoints. Fully endorsed by University of Cambridge International Examinations.
Author: Bettison, I
ISBN: 9780199138753
Published in 2011.
Published by Oxford University Press, UK More information on Extended Mathematics for Cambridge IGCSE - Teacher's Resource Kit (with CD) [New window]
Extended Mathematics for Cambridge IGCSE (with CD-ROM) Third Edition
The third edition of the bestselling Extended Mathematics for Cambridge IGCSE has been written for students following the University of Cambridge International Examinations syllabus for IGCSE Extended Mathematics. Written by a highly experienced author for the international classroom, this title covers all aspects of the syllabus content in an attractive and engaging format, and provides a wealth of support for students.
Author: Rayner, D
ISBN: 9780199138746
Published in 2011.
Edition: 3rd
Published by Oxford University Press, UK More information on Extended Mathematics for Cambridge IGCSE (with CD-ROM) Third Edition [New window]
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Barry Simon, I.B.M. Professor of Mathematics and Theoretical Physics at the California Institute of Technology, is the author of several books, including such classics as Methods of Mathematical Physics (with M. Reed) and Functional Integration and Quantum Physics. This new book, based on courses given at Princeton, Caltech, ETH-Zurich, and other universities, is an introductory textbook on representation theory. According to the author, "Two facets distinguish my approach. First, this book is relatively elementary, and second, while the bulk of the books on the subject is written from the point of view of an algebraist or a geometer, this book is written with an analytical flavor".
The exposition in the book centers around the study of representation of certain concrete classes of groups, including permutation groups and compact semisimple Lie groups. It culminates in the complete proof of the Weyl character formula for representations of compact Lie groups and the Frobenius formula for characters of permutation groups. Extremely well tailored both for a one-year course in representation theory and for independent study, this book is an excellent introduction to the subject which, according to the author, is unique in having "so much innate beauty so close to the surface".
Readership
Research mathematicians and graduate students.
Reviews
"Contains a very good explanation of representation theory of finite and compact groups and can be recommended to everyone for learning or teaching representation theory."
-- Zentralblatt MATH
"This is indeed a nice book ... I would recommend it precisely for the graduate course [that] I am teaching now, "Representation Theory" ... I very much like the hands-on approach and the very explicit formulae that are given ... Professor Simon has done an excellent job on this beautiful material."
-- Tudor Ratiu, University of California, Santa Cruz
"Can be recommended as a base for courses about representations of finite groups and finite-dimensional representations of Lie groups."
-- Mathematical Reviews
"This book closes a gap that impeded instruction in the principles of the theory of group representations in the past ... it would also tend to close the gap between the interested student of the subject and the advanced researcher keen on bare facts ... a most readable account of a subject of enduring fascination."
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This software to download was designed by a teacher to help teachers teach algebra I, algebra II, trigonometry, probability and statistics, and 3D graphing. It approaches these topics from a uniquely teacher point of view. For example, it generates problems for students to solve such as systems of equations that are independent-consistent, dependent-consistent, and inconsistent.inconsistent. It generates graphs that students must identify in function notation and/or by exact formula.
3D graphing techniques are illustrated dynamically. Polar and parametric equations can be investigated with the help of a "lightning bug". Slider graphs of any functions can be created to help students visualize the effects of parameters. Piecewise-defined functions and equations that do not represent functions can be graphed. Implicitly defined equations can be graphed. Inequalities can be graphed in standared or "reversed" mode
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Algebra 2 - 03 edition
Summary: Applications with "Real" Data Since the graphics calculator is recommended, students experience excitement as they use "real" data in Algebra 2. Students investigate and extend relevant applications through engaging activities, examples, and exercises. Graphics Calculator Technology In Algebra 2, the graphics calculator is an integral tool for presenting, understanding, and reinforcing concepts. To assist student...show mores in using this tool, a detailed keystroke guide is provided for each example and activity at the end of each chapter. Functional Approach Algebra 2 examines functions through multiple representations, such as graphs, tables, and symbolic notation. Working with transformations (investigating how functions are related to each other and their parent functions) prepares students for advanced courses in mathematics by developing an extensive, workable knowledge of functions0030660542-5-0
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The vision of the mathematics standards is focused on achieving one central goal: to enable ALL of New Jersey's children to acquire the mathematical skills, understandings and attitudes that they need to be successful in their careers and daily lives.
Students at JohnP.StevensHigh School are required to successfully complete a mathematics course in grades 9, 10, 11 and 12.Student placement is based upon aptitude, district criteria and classroom performance.
Department offerings are available in the district's program of studies.All courses are aligned with the New Jersey Core Curriculum Standards for Mathematics.
Mathematics teachers at John P. Stevens are the department's greatest resource.Besides an excellent and thorough knowledge of the subject, each teacher utilizes differentiated instructional strategies that enhance student success.In-service programs and workshops are conducted throughout the year to assure professional growth for all of our teachers.
The John P. Stevens Mathematics Department focuses not only on the content of the curriculum but it's application to real-life situations.Our students will always be prepared for the challenges they will face during their college experience.Calculators, computers, manipulatives, technology and the Internet are used as tools to enhance learning and assist in problem solving.
In this changing world students who have a good understanding of mathematics will have many opportunities throughout their lives.The John P. Stevens Mathematics Department is committed to providing all students with the opportunity and support necessary to learn and understand significant mathematics.
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High School Advantage 2008 was specially developed to supplement classroom
curriculum by including award-winning content that support state standards.
In-depth lessons with sample problems and questions teach, reinforce and track
student progress in the core subjects.
A Complete Student Resource Center now on DVD with 10 core subjects along with
after school extras including PC games and more!
Skills Learned
Algebra II
Geometry and Trigonometry
Composition
Economics
World History
U.S. Government
Foreign Language
Typing
Biology
Chemistry and Physics
Product Features
10 Core Subjects Expanding on Key Academic Areas
1,600+ Lessons Deliver Easy-to-Understand Concepts and Tutorials
2,000+ Exercises Adapt to Different Learning Levels and Styles
Supports State Standards
Student Planner
Study Tools
After School Extras for today's students include music, PC games, mobile
games, ringtones
Educational Features
Mathematics Over 265 Lessons and 900 Exercises Solidify critical math skills and prepare for College exams or professional
endeavors with lessons and exercises in the key subject areas.
Algebra II
Roots
Functions
Conic Sections
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Geometry and Trigonometry
Triangles
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Reasoning and Equality
Coordinate Geometry
Graphing Sine and Cosine
Trigonometric Functions
Trigonometric Equations
Economics
Practice micro-economics in a simulated environment with 1,000 different company
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Observe how markets function in a dynamic simulated setting.
Compete in an interactive game that allows you to see the impact of
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Students are asked to manage a variety of companies in different market
types ranging from single company monopolies to highly competitive markets.
Science Over 180 Lessons and 800 Animations Supplement classroom experience and review Biology, Chemistry and Physics at
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Biology
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History and U.S. Government Over 300 Lessons and 75 Video Clips Students retrace historical events and review key concepts in American
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Just the math skills you need to excel in the study or practice of engineering
Good math skills are indispensable for all engineers regardless of their specialty, yet only a relatively small portion of the math that engineering students study in college mathematics courses is used on a frequent basis in the study or practice of engineering. That's why Essential Math Skills for Engineers focuses on only these few critically essential math skills that students need in order to advance in their engineering studies and excel in engineering practice.
Essential Math Skills for Engineers features concise, easy-to-follow explanations that quickly bring readers up to speed on all the essential core math skills used in the daily study and practice of engineering. These fundamental and essential skills are logically grouped into categories that make them easy to learn while also promoting their long-term retention. Among the key areas covered are:
With the thorough understanding of essential math skills gained from this text, readers will have mastered a key component of the knowledge needed to become successful students of engineering. In addition, this text is highly recommended for practicing engineers who want to refresh their math skills in order to tackle problems in engineering with confidence.
The EPUB format of this title may not be compatible for use on all handheld devices.
Details
ISBN: 9781118211106
Publisher: John Wiley & Sons, Ltd.
Imprint: Wiley-IEEE Press
Date: Sept 2011
Creators
Author: Clayton R. Paul
Reviews
"Summarizing, this is a very nice textbook, covering many interesting topics and written in a very digestible manner, which can be warmly recommended to students in natural sciences, computer science, and all branches of engineering." -
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A rigorous, concise development of the concepts of modern matrix structural analysis, with particular emphasis on the techniques and methods that form the basis of the finite element method. All relevant concepts are presented in the context of two-dimensional (planar) structures composed of bar (truss) and beam (frame) elements, together with simple discrete axial, shear and moment resisting spring elements. The book requires only some basic knowledge of matrix algebra and fundamentals of strength of materials.
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In my opinion, one of the most important concepts to discuss in a liberal arts math course is the notion of mathematical proof—what it is, why mathematicians put so much emphasis on it, whether it is overrated, and how the concept has evolved over time. There are several ways to approach this subject.
Give examples of proofs. This MO question as well as this one provide some nice examples. I'd also recommend the MAA series of books on Proofs Without Words.
Discuss the role of computers and experiment in mathematics. Jonathan Borwein has co-authored several books on experimental mathematics, e.g,. The Computer as Crucible. Though much of the mathematical content may be too advanced, the introductions to these books are extremely lucid and valuable.
Discuss the foundations of mathematics. My top recommendation in this category would be Torkel Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Again, some sections of the book may be too technical, but there are plenty of extremely well-written and valuable non-technical sections.
Even among the educated public, one frequently encounters people who have no concept of how unique mathematical proof is, who think that computers have put mathematicians out of business, and who have heard just enough about Gödel's Theorem to be dangerous. The above readings should go a long way towards dispelling these common misconceptions.
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Math
Math is an important subject in the context of high school education. If the students are able to succeed in this subject then the overall success in the high school curriculum would be relatively easy. Perhaps, the subject is more important in the long term running. Having a good concept in the math curriculum of high school standards would prepare the learners well for college and career. Education Services Aug offers online math solutions that are very engaging and interesting, which enables the high school learners to succeed in math whether it be in algebra or calculus. Our math curriculum is aligned to the standard that is followed in the courses recognized in the United States.
The basic structure of the coursework is to motivate the students and generate interests in the subject. We encourage the students in various problem solving standards. It is important to read the problems and understand them. There should be perseverance on their part in solving these problems. The learners should be able to reason conceptually and express them quantitatively. Arguments against each problem should be viable and made from the understanding of the concepts. It is also healthy to be critical of other's arguments as well. Our interactive learning process provides the students with modeling and visualizing techniques for mathematics. The learners are familiarized with the use of appropriate tools at strategic moments. Education Services Aug secondary math solutions instill in the learners the importance of attending to precision. The students are taught to look for and use the structures in the problems and also to express regularity in reasoning again and again.
Our comprehensive and flexible online courses allow the students to learn at the pace that is best suited to them. We offer an extensive coverage of the course structure. Education Services Aug math coursework includes Advanced Calculus, Algebra, Consumer Mathematics, Pre-Algebra, Pre-Calculus and Geometry. Apart from that we also offer many skill based packages like Data Skills, Foundation Mathematics, Math Problem solving and Trigonometry skills packages. Based on this programs students are taught to think independently and intuitively. The aim is to make math an easy subject so that students can relate to it and not feel threatened by it. Education Services Aug also provides assessments in order to have an idea about the gaps in the understanding of the concepts of each learner which would benefit the educators to channelize them in the correct path to success.
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Courses in
Mathematics
and Statistics
Honours
timetable
MT2003 APPLIED MATHEMATICS
Aims
To introduce students to applied mathematics through the construction, analysis and interpretation of mathematical models for problems arising in the natural sciences, and to introduce students to the techniques of analysis used in mathematical modelling, such as numerical methods, dimensional analysis, solution of ordinary and partial differential equations.
Mathematical models from population dynamics, Newtonian dynamics and wave motion that lead to ordinary and partial differential equations are developed in detail, and the basic elements of vector calculus in three dimensions are introduced.
Objectives
By the end of the course the students should be familiar with:
- Kinematics and the vector formulation of Newton's Laws
- Newtonian model of gravity, motion in constant gravity and particle motion under a variable force
- The energy equation as first integral of equation of motion, the concepts of kinetic and potential energy
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A comprehensive textbook covering not only the ordinary theory of the deformation of solids, but also some topics not usually found in textbooks on the subject, such as thermal conduction and viscosity in solids. more...
The learn-by-doing way to master Trigonometry Why CliffsStudySolver Guides? Go with the name you know and trust Get the information you need--fast! Written by teachers and educational specialists Get the concise review materials and practice you need to learn Trigonometry, including: Explanations of All Elements and Principles * Angles and quadrants... more...
Though the Japanese Abacus may appear mysterious or even primitive to those raised in the age of pocket calculators and desktop computers, this intriguing tool is capable of amazing speed and accuracy. It is still widely used throughout the shops and markets of Asia, and its popularity shows no sign of decline. Here for the first time in English is... more...
How Chinese Teach Mathematics and Improve Teaching builds upon existing studies to examine mathematics classroom instruction in China. It combines contributions from Chinese scholars with commentary from key Western scholars to offer multiple perspectives in viewing and learning about some important and distinctive features of mathematics classroom... more...
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Practice Makes Perfect has established itself as a reliable practical workbook series in the language-learning category. Now, with Practice Makes Perfect: Calculus, students will enjoy the same clear, concise...
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In its largest aspect, the calculus functions as a celestial measuring tape, able to order the infinite expanse of the universe. Time and space are given names, points, and limits; seemingly intractable problems...
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Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern...
$ 14.79Tough Test Questions? Missed Lectures? Not Enough Time?Fortunately for you, there's Schaum's.More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is...
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Many colleges and universities require students to take at least one math course, and Calculus I is often the chosen option. Calculus Essentials For Dummies provides explanations of key concepts for students...
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Math e-Books for $0
[29 Aug 2011]
Most of the following free (or low cost) math e-books are PDF versions of ordinary math books.
You probably won't find your assigned text book here, but you'll find something that is pretty close. And for the millions of keen students who cannot afford the high price of math text books, this will be a valuable list.
Copyright information: It's not clear if copyright permission has been granted in some of these collections. In some cases, the business model involves advertising throughout the book (but the quality tends to be higher). In Google Books' case, for many of the books, they've been given permission to show selected pages only.
Google Books
Google wanted to digitize every book in the world, but not surprisingly, they ran up against copyright issues. Many of these books are not complete, but can still be very useful for that nugget of information you're looking for.
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Based on Essential Skills foundations and developed by experienced training professionals, this workbook is designed to help instructors develop math worksheets for apprentices in technical training settings. Users learn how to incorporate Essential Skills into worksheets that will meet instructional goals and help apprentices to learn essential trades math.
The authors point out that during training and on a work site, apprentices need more than just math skills. They need strong Essential Skills (ES) to succeed in training and on the job. This guide can help instructors learn how to apply ES understanding to the task of making worksheets that meet instructional goals and help apprentices learn trades math more effectively.
You can purchase a hard copy of this document on the Construction Sector Council's website at
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Calculus is often a difficult subject for students. They are forced to look at familiar mathematical objects from a radically different perspective, and perform computations much more intricate than anything experienced before. Fortunately, with proper guidance calculus can be made bearable; indeed, after a while it starts to feel procedural!
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Next year I will be starting a degree in game programming. I have been told to brush up on my math. Which chapters of math text books should I read over?
The course is described as:
The Qantm Bachelor of Interactive
Entertainment (with a Major in Games
Programming) focuses on specific areas which
are critical to developing knowledge
and skills in games programming for
interactive entertainment. Students of
the Major in Games Programming learn
C++ programming, work with
mathematical functions and artificial
intelligence to design and program
games for a variety of devices. To
broaden students' perspectives on the
games development pipeline, the major
includes specialised courses in script
writing, character development, games
design, agent systems and 2D and 3D
animation.
While, the topics are similar, there a differences, the inclusion of a course description allows for more distinct answers relevant to my unique situation. – fauxCoderOct 19 '09 at 1:09
@Ngu Soon Hui, this is not a duplicate of that question. I'd say that the other questions is potentially a subset of this one. For example, path finding requires knowledge of graph theory. – Bob CrossOct 19 '09 at 1:32
@cdiggins, why lambda calculus? This is a serious question: I don't understand how this would be useful to someone who was specifically writing games. – Bob CrossOct 19 '09 at 12:50
LC is useful to understand the underpinnings of functional programming. I think all programmers should know it. Then again, one can do what I did, and learn it after learning functional programming in practice. – cdigginsOct 20 '09 at 3:14
Having studied that course at Qantm a few years back ('06 graduate), I know approximately what you're in for ;)
All the maths you will need will be taught as part of the course - however it is done at a blazing pace, and if you miss any of the tutorials you will need to scramble and get your head around it yourself. I remember missing the class on collision detection, and boy was that painful.
All the answers given already are spot-on for the useful concepts for game math. One extra thing you might want to get on top of early is state machines (for AI).
Although this article's audience is someone who is preparing for an interview, I still believe the first section in this article (Math) gives you an idea of what you should know, or at least be comfortable with.
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analyze implicit differentiation using technology. In this calculus lesson, students solve functions dealing with implicit differentiation on the TI using specific keys. They explore the correct form to solve these equations.
Twelfth graders solve first order differential equations using the separation of variables technique. In this calculus instructional activity, 12th graders explain the connection between math and engineering. They brainstorm what engineers do in real life.
In this calculus worksheet, students perform implicit differentiation to take the derivative. They solve function explicitly of x. They differentiate between implicitly and explicitly. There are 16 problems with an answer key.
Students calculate the maxima and minima of quadratic equations. In this calculus instructional activity, students apply the derivatives by finding the maxima and minima using real life application. They solve optimization using the derivative.
Students solve quadratic equations using calculus. In this optimization lesson, students create parabolas using SolidWorks and use it to identify the maximum and minimum point on their function. This assignment is done online and with pencil and paper.
Students derive functions given a limit. In this calculus lesson, student define the derivative of f at x=a, knowing the derivative is a point or just a number. This assignment requires students to work independently as much as possible.
Students discuss the following topics of Calculus: The Tangent Line Problem, The Area Problem, and Exercises. They find limits graphically and numerically. Students write a mathematical autobiography, they write their earliest memories of mathematics or numbers.
Students investigate the capabilities of the TI-89 calculator. In this Calculus lesson, students explore the statistical, graphical and symbolic capabilities of the TI-89. Students investigate topics such as solving systems of equations, finding inverse functions, summations, parametrics and trigonometry.
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Cartoon Guide To Calculus - 12 edition
Summary: A complete---and completely enjoyable---new illustrated guide to calculus Master cartoonist Larry Gonick has already given readers the history of the world in cartoon form. Now, Gonick, a Harvard-trained mathematician, offers a comprehensive and up-to-date illustrated course in first-year calculus that demystifies the world of functions, limits, derivatives, and integrals. Using clear and helpful graphics---and delightful humor to lighten what is frequently a tough subject---he teach...show morees all of the essentials, with numerous examples and problem sets. For the curious and confused alike, The Cartoon Guide to Calculus is the perfect combination of entertainment and education---a valuable supplement for any student, teacher, parent, or professional
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Book Description: Teacher's Edition. This edition balances the investigative approach that is the heart of the Discovering Mathematics series with an emphasis on developing students' ability to reason deductively. If you are familiar with earlier editions of Discovering Geometry, you'll still find the original and hallmark features, plus improvements based on feedback from many of your colleagues in geometry classes.
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Working with charts, graphs and tables
Your course might not include any maths or technical content but, at... you how to use charts, graphs and tables to present your own information.
After studying this unit:
you will learn how to reflect on your mathematical history and existing skills, set up strategies to cope with mathematics and assess which areas need improving;
through instruction, worked examples and practice activities, you will gain an understanding of the following mathematical concepts:
reflecting on mathematics,
reading articles for mathematical information,
making sense of data,
interpreting graphs and charts;
you will be provided with a technical glossary, plus a list of references to further reading and sources of help, which can help you improve your maths skills.
Working with charts, graphs and tables
Introduction openlearn unit LDT_4 More working with charts, graphs and tables, which looks into more ways to present statistical information and shows you how to use charts, graphs and tables to present your own information.
Comments
This might sound stupid but i am trying out a free maths intro...
Trouble is, the articles we need to read and review are so small. How do i zoom out so that i can read the article ???
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Our users:
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My son used to hate algebra. Since I have purchased this software, it has surprisingly turned him to an avid math lover. All credit goes to Algebrator04-08:
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Algebra and Trigonometry With Analytic Geometry: series of highly respected Swokowski/Cole mathematics texts The goal of this text is to prepare students for further courses in mathematics. This book is set apart from the competition in a number of ways: it is mathematically sound, it focuses on preparing students for further courses in mathematics, and it has excellent problem sets. This edition has been improved in many respects. All of the chapters include numerous technology inserts with specific keystrokes for the TI-83 Plus and the TI-86, ideal for students who are working with a calculator for the first time. The new design of the text makes the technology inserts easily identifiable, so if a professor prefers to skip these sections it is simple to do so.
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Introduction to Economic Analysis This book presents standard intermediate microeconomics material and some material that, in the authors' view, ought to be standard but is not. Introductory economics material is integrated. Standard mathematical tools, including calculus, are used throughout. The book easily serves as an intermediate microeconomics text, and can be used for a relatively sophisticated undergraduate who has not taken a basic university course in economics.
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Algebra InterMath is a professional development effort designed to support teachers in becoming better mathematics educators. It focuses on building teachers' mathematical content knowledge through mathematical investigations that are supported by technology. InterMath includes a workshop component and materials to support instructors. ForFourier: Making Waves Learn how to make waves of all different shapes by adding up sines or cosines. Make waves in space and time and measure their wavelengths and periods. See how changing the amplitudes of different harmonics changes the waves. Compare different mathematical expressions for your waves.Coordination problem Coordination problem is a blog produced by a team of US based professional economists from the Austrian school commenting on scholarly research in economics and current events. It promotes the work of Hayek, Mises, Kirzner and others of the Austrian school - a non-mainstream (heterodox) schools of economists, who advocate that the complexities of human economic behaviour makes the mathematical modelling of markets extremely difficult. The blog features links to related blogs, internet websites a Author(s): No creator set
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Symmetry Weaths everywhereStarting with maths: Patterns and formulasalysing skid marks This unit is the second in the MSXR209 series of five units on mathematical modelling. In this unit you are asked to relate the stages of the mathematical modelling process to a previously formulated mathematical model. This example, that of skid mark produced by vehicle tyres, is typical of accounts of modelling that you may see in books, or produced in the workplace. The aim of this unit is to help you to draw out and to clarify mathematical modelling ideas by considering the example. It assumMathematical language In our everyday lives we use we use language to develop ideas and to communicate them to other people. In this unit we examine ways in which language is adapted to express mathematical An unsolved problem! Inequalities
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The reviews below refer to free (or free-to-try)
off-site tutoring and instructional resources. To access the Purplemath lessons and tutoring forums,
please use the links to the right. For paid in-home tutoring, please try here.
algebra.help: This site
has lessons on basic algebra topics and techniques, study tips, calculator advice, worksheets,
and more.
BestDamnTutoring.com:
In contrast to the YouTube norm, this tutoring crew directed their algebra instructor through multiple
"takes" to ensure clarity; the videos are generally
short, to the point, and error-free.
Brightstorm:
This video compendium offers videos on many topics, such as chemistry, calculus, and ACT test-prep.
In particular, you will find a large collection of algebra lessons.
CliffsNotes:
This recognized name in helpful supplementary resources has added online lessons to its print offerings.
There are loads of math lessons, including many for algebra.
Exercises
in Math Readiness: EMR has lessons, examples, and short quizzes (complete with hints and solutions).
They cover only a few topics, but the coverage is excellent, and extends from algebra to trigonometry and set theory.
FreeMathHelp:FreeMathHelp
has some lessons covering various topics from algebra to calculus, a worksheet generator, and a
message board which offers free tutoring. Registration
for the tutoring forum is required, but is free and fast. Questions are usually answered within
a day. For math formatting advice, follow the links in the "Forum Help" pull-down
menu at the top of every forum page.
Joseph Coffman's Lecture
Notes: Mr. Coffman's lectures cover a
lot of material and include many worked examples. The notes center on algebra, but also include a
little statistics (box-and-whisker plots, for example) and trigonometry. Each lesson is linked
to the related Glencoe online learning resource.
Karl's Notes on Email:
You may have noticed that it's hard to write
out math problems when all you have is your e-mailer to work with. Because of this, math people
have developed commonly accepted ways of formatting math for the purposes of e-mail and newsgroups,
some of which has been adopted into (or from) the syntax used by graphing calculators. Karl's Notes are an excellent overview of this
formatting.
Khan Academy:
If you're tired of doing searches trying
to find algebra videos on various different topics, you can now start with an extensive listing
in one place. Salman Khan has loads of videos
teaching algebra and other topics.
MathCelebrity:Don Sevcik has created an extensive set of online step-by-step
solvers. If you find it helpful to see the steps, so you can learn how to do the rest of the exercises
for yourself, these javascript solvers might
be just the thing. (No installation or plug-ins required.)
Mathnerds: Once you've registered
(membership is free), log in to this
tutoring site. Then pick the category that most closely matches what you are studying and submit
your question. It will be assigned to a qualified tutor. Questions are answered by pre-qualified
tutors, usually within a day or two. Valid e-mail address required.
MathOps: This site is meant for teachers
and classrooms, but there is loads of great free material, too. From the home
page, click the link for "Free Lessons". (To return to their home page, you'll need
to use your browser's "Back" button, or scroll down to the bottom for a link.)
Maths Is Fun: If you'd like
extra practice or instruction on pre-algebra or early-algebra topics, Maths
Is Fun is a great resource. The site also has worksheets, a tutoring forum, puzzles, and
teaching games.
One Mathematical
Cat:
Professor Fisher has created entire textbooks and posted them online. The algebra
text includes "Web Exercises" which you can use for practice. (Be sure to read their
instructions.)
OpenStudy: This free "groups"
site offers an interface for posting (and answering) questions in math
and other topics.
Paul's Online Math Notes: Paul Dawkins
of Lamar University has compiled some very nice lessons, reviews, and cheat-sheets for his college students,
and has made his materials available to the rest of us, too. His site
covers algebra through differential equations. The lessons are very thorough, with lots of worked
examples, sensible advice regarding common mistakes, and helpful previews of what to expect in
later courses.
Professor Kuniyuki's Precalculus Notes:
Professor Kuniyuki's advanced algebra notes are
keyed to a particular textbook, but, being listed by topic, anybody can use them. His lessons
are in PDF form and tend to be somewhat technical (textbook-ish), but the advice and warnings they
contain are very good.
Professor Symancyk's
algebra lessons: Professor Symancyk has
written some great lessons, which include illustrations and worked examples. Pick your topic from
his menu. (Note: His e-mail
is for his Maryland [USA] students only.)
Regents Prep: The Oswego
City School District provides many lessons covering different topics
for many grade-levels. Of interest to the algebra student are the lessons designed for
the Regents Prep exam, many of which contain instructions related to graphing calculators. Scroll
down on the "Math A" and "Math B" pages to view each index of lessons.
Stan Brown's Math and Calculator articles: Professor Brown has created a nice collection of tutorials
covering many common tasks, and some not-so-common ones, for classes from algebra through calculus
and statistics. Includes programs you can download and install, step-by-step instructions, illustrations,
and a conversational tone.
University of Arizona
Software: This software contains self-testing quizzes, but the "Help" contains good
lessons. The programs are DOS-based, but VERY user-friendly. Scroll down the page to "Are
You Ready?", and choose your level.
WTAMU Virtual Math Lab: The West Texas A&M University's Virtual
Math Lab has a lengthy list of tutorials, covering topics throughout algebra. Each lesson includes
useful terminology, worked examples, and links to other sites.
WyzAnt: This online
tutoring service also offers a long list of math lessons, including
algebra and pre-calculus.
xyAlgebra:
If you are having trouble with beginning algebra, especially word problems, this free package
may be just the thing. The software shows all of the steps and reasoning for doing basic algebra
problems, and allows the student to work through exercises, providing lessons on necessary background
topics, as needed.
If you think your site should be listed here, please submit
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Math 12 - Part 2 (MATH 0993)
This course is the second half of ABE Provincial Level (Algebra and Trigonometry) Mathematics, which prepares students with the algebra and trigonometry skills needed for post-secondary programs that require Math 12 equivalency. It includes: trigonometric functions and their inverses, graphs of circular functions, trigonometric identities and equations, solving triangles using sine and cosine laws, systems of equations, conic sections, sequences and series. Both MATH 0983 and MATH 0993 are required for Grade 12 Provincial Level (Algebra and Trigonometry) Mathematics equivalency.
Prerequisite:MATH 0983 with a C- (Math 12-Precalculus with a C-, or Math 12-Principles with a C-, or equivalent).
Course Outline: 53 KB
Credits:
0.0
Programs offering this course:
To register for this course, you must apply to the college and be accepted into one of the following programs:
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Calculators Are for Calculating, Mathematica Is for Calculus
Andy Dorsett
In this Wolfram Mathematica Virtual Conference 2011 course, learn different ways to use Mathematica to enhance your calculus class, such as using interactive models and connecting calculus to the real world with built-in datasets.
See how Wolfram technologies like Mathematica and Wolfram|Alpha enhance math education. The video features visual examples of course materials, apps, and other resources to help teachers and students cover math from algebra to calculus to statistics and beyond.
Watch an introduction to the Wolfram Demonstrations Project, a free resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and a range of other fieldsIn this video, get a quick introduction to the Wolfram Education Portal, which features teaching and learning tools created with Mathematica and Wolfram|Alpha, including a dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.
computerbasedmath.org is a project to build a new math curriculum with computer-based computation at its heart. In this talk from the Wolfram Technology Conference 2011, Conrad Wolfram discusses the concept, progress, and plans.
Explore the various ways mobile devices can tap into the power of Mathematica and Wolfram|Alpha to enhance learning in math, science, and even music classrooms in this recorded presentation from the Wolfram Technology Conference 2011.
In this Wolfram Mathematica Virtual Conference 2011 course, learn different ways to use Mathematica to enhance your calculus class, such as using interactive models and connecting calculus to the real world with built-in datasets Portuguese audio.
Mathematica can be used to enhance course management systems by helping teachers easily communicate ideas, give students immediate feedback, and link real-world datasets to textbook examples. Learn more in this screencast.
This is the third video in a series showing examples of Mathematica that are features especially useful for K–12 and community college educators. Topics include mathematical typesetting, slide shows, interactive models, and more.
This is the second video in a series showing examples of Mathematica features that are especially useful for K–12 and community college educators. In this video, you'll discover how easy it is to create interactive Demonstrations, lessons, quizzes, and instructional handouts with Mathematica.
This is the first video in a series showing examples of Mathematica features that are especially useful for K–12 and community college educators. In this video, educators share firsthand experiences of teaching with Mathematica Spanish audio.
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This workshop is designed for teachers who are new to the IBMaths Studies course and aims to help delegates familiarise themselves with the make up and philosophy of the course. ... For a good textbook treatment of . conditional probability. and .
IB-produced teacher support material (TSM) ... from their own and other subject groups teachers working in isolation multiple sources and resources for learning a textbook-driven curriculum students investigating, questioning, ...
TEXTBOOK: Mathematics Studies SL by Robert Haese and Mal Coad. ICT: Powerpoint. Number and Algebra. ... Every IB course should contribute to the development of international mindedness in students.
In particular we are an International Baccalaureate world school. ... and the students start to use their GCSE textbook in Year 9. ... Mathematics is a very popular and successful subject in our Sixth Form with two thirds of our students taking A level Maths, Further Maths or IB options.
International Baccalaureate. Primary Years Programme. ... A choice of Chinese Maths or English Maths. ... PYP students access a wide range of learning resources and materials and do not follow one specific textbook. ( Collaborative learning.
[See also holdings for newer editions, if any]. Textbook recommended for a few of our courses/modules. May be out of print now. MMM114 Materials Science I. (a+, Oct) check on MMM Book List. MMM211 Materials Science ... ENM101 Engineering MathsIB. (a, Oct) check on ENM Book List. and later modules.
In IB Chemistry we will use the ... site about the ozone hole. Chem101 Chemdex Sheffield University's directory of Chemistry on the Internet ... a review' in the section 'Maths and ...
A textbook will be used to resource the section on the French Revolution. Ancient Civilizations: ... The IB system will also be looked at online so that students come to an understanding of the IB system. Subject: Maths Grade: 6i Term: ...
ib_size=0.260000. ib_r=0.250000. ib_g=0.250000. ib_b=0.250000. ... All of this will appear in a decent mathstextbook. Obviously, you don't need to know all of this for most cases - it's a very rare preset that uses more than one or two of these curves.
The information contained in these notes is correct to my knowledge and by no means copied from any textbook, therefore, ... Ie = Ic + Ib (1) i.e. Current flowing into the ... This time I'll go through the maths a lot faster, and let you work it out. ie = ib + (ib hence, Vi = ib.r ...
Basic maths: introductory course. Resultados de Aprendizagem: ... The preferred ones are marked with an *. 1 textbook of International Business among the following: Daniels, John D. and Lee H. Radebaugh "International Business.
Should I Do More to Upgrade My Maths? What Study Skills Will I Need? ... e.g. IB) you will be taking ... For the revision of basic algebra, any GCSE (Higher Level) textbook will be useful, although Part 1 (chapters 1 to 5) of G. Renshaw. 2005. Maths for Economics, ...
There exist strong beliefs among them that if you do not follow the textbook, ... at the same time as some have great ambitions and want to enter the International Baccalaureate (IB) programme. ... This is more maths. Before, it was almost as kindergarten.
Elements was widely considered the most successful and influential Mathematics textbook of all time, ... A complex number a + ib represents a point in a plane. ... Russian maths genius Perelman urged to take $1m prize bbc.co.uk, Wednesday, 24 March 20102.
The history of maths is a serious part of the study of Mathematics in Europe, ... This was written by a Victorian teacher who has worked with the IB structure: "In Victoria, ... and textbook and other resource developers will 'do their own thing", ...
... the IB Middle Years and Diploma programmes and the Pre-U examination. ... when compared to other conventionally 'hard' subjects such as maths and physics; ... or superior to the exemplar material in the course textbook.
The textbook and the curriculum drive goals and objectives with some attention to students' needs and achievement. ... IB, IC. 3. What teaching strategies will you use to teach this lesson? IB, IIA, IIB. 4. How will you assess student learning? Identify specific data. IB, IC, IIIA, IIIC. 5.
Introduction of IB (School & History Group Dev Plan) Review KS4. ... the History curriculum prepares the students to perform basic tasks such as how to use the index and glossary of a textbook. ... Maths Lesson; Racial Instruction; History lesson; Look at school life in Nazi Germany. Lesson 3
Maths. Music. Physical Education. Physics *Spanish (Language B2) Technology (ICT) ... IB information will also be offered in handout form so students come to an understanding of the IB system. ... Tasks from within the textbook.
The International Business course (BSB50807) and the Diploma of Accounting ... Shepherd, J. – Business Maths and Statistics, 2nd Edition 2007 Learn Now Pty Ltd OSP107 ... Textbook received Unit outline received Student kit received 4 weeks after commencement of course/semester ...
Tuition fees, PTA fees, and textbook costs were very low for the bottom four quintiles. The main expenditure was on educational materials other than textbooks. In contrast, in 1995, monthly expenditure fees ranged from $0.82 for the lowest quintile to $2.67 for the richest quintile.
can you unambiguously write down what LCP states (see 148 of the AS textbook)? remember this is a . predictive. ... Maths tip: Any number raised to the power zero = 1. Don't panic !! – this can be explained easily in words – ask your teacher.
... asked the class which of us enjoyed Maths and preferred translating from English to Latin rather than from Latin ... I had been amazed by a sentence in my history textbook mentioning Julius Caesar's De Bello Gallico as something `you ... But this is an insidious and damaging course.(ib.).
... reading is recommended in the form of a reader that will be supplied with the study material or a prescribed textbook that should be bought by the ... aspects covered in the business plan, international business plan, a pro-forma business plan; Basic financial planning – basic ...
1 Carnegie unit in an IB course with a score of 4 or higher on the exam OR. ... (English I, English II, 2 required maths, 2 required sciences, 2 required social studies ... and current events related to astronomy. The course work will include textbook assignments, Internet activities, online ...
Throwing out the textbook 1 18. by Ana Coll and Luis Fernandez. Teacher or syllabus designer? 22. by Pilar Romera. ... Ana Coll is a secondary school teacher of English at IB Juan de Austria, Barcelona, Spain. She has got an MA in Linguistics and ELT, University of Leeds.
In fact many Maths staff have been taught about the use and construction of triangular graphs in Geography. ... rather than working purely from the textbook. ... As our school considers the International Baccalaureate it becomes clear that this approach is the IB approach ...
THE PROFILE OF OUR SCHOOL IS MATHS-SCIENCES OVER ALL THE FOURTH GRADES. Several times, with several persons ... More of Particle Physics I teach within IB Diploma Programme. ... A textbook with Q/A about particle physics. 36 Petr Jílek. 5 - 10 times.
... science instruction typically occurred as reading the science textbook and answering ... of these three teams, two serve the general populations and one serves the students in the Pre-IB program ... (Norway); CETL-MSOR Conference 2008: Shaping the Future of Maths & Stats in ...
Maths) (from March) Ms Jill ... is in his fourth year of a five-year appointment as Chief Examiner for Mathematical Studies with the International Baccalaureate ... to the book: M Ryan (ed), Jewish–Christian Relations: a textbook for Australian students (Melbourne: David Lovell ...
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$81.90 9/1 book offers a "six-step approach" to problem solving, numerous tips, and clear, concise explanations that explore the concepts behind mathematical processes. Simplified language appeals to a variety of learning styles, and promotes active, independent, and lifelong learning -- while strengthening critical thinking and writing skills. This book addresses curriculum and pedagogy standards that are initiatives of the American Mathematical Association of two-year Colleges (AMATYC), the National Council of Teachers of Mathematics (NCTM), and the Mathematics Association of America (MAA). It focuses on number, symbol, spatial and geometric, function, and probability and statistical sense. Other features include career applications, and mathematics in the workplace articles that demonstrate the relationship of chapter concepts to highly sought after job skills -- such as computational, research, and critical thinking/decision-making.
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Introduction to Mathematical Reasoning
The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems. Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations. The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except the first one, puzzles. There are several reasons for including puzzles. First and foremost, a challenging puzzle can be a microcosm of mathematical development. A great puzzle is like a laboratory for proving propositions. The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem. The mathematician "plays" with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler "plays" with the puzzle. Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing. At some stage, the puzzler (mathematician) develops sufficient sense of the structure and only then can he begin to build the solution (prove the theorem). This multi-step process is perfectly mirrored in solving the KenKen problems this course presents. Some aspects of the solutions motivate ideas you will encounter later in the course. For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity.
Requirements for Completion: In order to complete this course, you need to work through each unit and all of its assigned materials. Pay special attention to Units 1 and 2 as these lay the groundwork for understanding the more advanced, exploratory material presented in latter units. You will also need to complete:
Subunit 1.2 Activity
Subunit 1.3 Activity
Sub-subunit 1.4.1 Activity
Sub-subunit 1.4.2 Activity
Unit 1 Assessments
Sub-subunit 3.4.5.1 Activity
Sub-subunit 3.4.5.2 Activity
Sub-subunit 3.7.1 Activity
The Final Exam
Note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through the assignments and assessment 112.25 approximately 31.5 hours to complete. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 4 hours) on Monday night; subunit 1.2, which is optional (a total of 4 hours) on Tuesday night; subunits 1.3 and 1.4 (a total of 4.25 hours) on Wednesday night; etc.
Tips/Suggestions: As you read, take careful notes on a separate sheet of paper. Mark down any important equations, formulas, definitions, and proofs that stand out to you. These notes will be useful to review as you study for the Final ExamIn this unit, you will begin by considering various puzzles, including Ken-Ken and Sudoku. You will learn the importance of tenacity in approaching mathematical problems including puzzles and brain teasers. You will also learn why giving names to mathematical ideas will enable you to think more effectively about concepts that are built upon several ideas. Then, you will learn that propositions are (English) sentences whose truth value can be established. You will see examples of self-referencing sentences which are not propositions. You will learn how to combine propositions to build compound ones and then how to determine the truth value of a compound proposition in terms of its component propositions. Then, you will learn about predicates, which are functions from a collection of objects to a collection of propositions, and how to quantify predicates. Finally, you will study several methods of proof including proof by contradiction, proof by complete enumeration, etc.
Instructions: Please click on the link above and read the entire article. Try not to get sidetracked looking at variations. Pay special attention to the growth of the number of Latin squares as the size increases. Note that if you want to look ahead at the type of problem you will be asked to solve, check the file "Logic.pdf" at the end of Unit 2.
Instructions: Please click on the link above. Scroll down the webpage to "Games," and select the "Sudoku" link to download the PDF. Read Tom Davis' paper, paying special attention to the way he names the cells and to his development of language. Next, if you have not done Sudoku puzzles before, Web Sudoku and Daily Sudoku and are two popular sites. Do one or two before moving on to Ken-Ken. Please note that this reading also covers the topics outlined in sub-subunits 2.1.2 and 2.1.3.
This will take you about 3 hours if you have not done Sudoku before, and about 2 hours if you have.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
Instructions: Please keep in mind that this activity is optional. After reading Harold et al.'s paper, click on the link above to access Ken-Ken puzzles, and attempt to complete one of these puzzles. Note that you can choose the level of difficulty (easier, medium, and harder). After a few practices, challenge yourself to attempt a Ken-Ken puzzle that is at the next level of difficulty. Do not allow yourself to get addicted!
You should dedicate no more than 1 hour to practicing Ken-Ken puzzles.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: This article is optional. If you have an interest in solving Ken-Ken problems, then you will find this section interesting. Otherwise, omitting it will not hinder your understanding of subsequent material. Should you choose to work through this section, please click on the link above, and read the paper by Harold Reiter, et al. for an introduction to Ken-Ken. Complete the exercises in the PDF. After you have completed the exercises, check Harold Reiter and John Thornton's "Solutions to Using Ken-Ken to Build Reasoning Skills." Please note that this reading covers the topics outlined in sub-subunits 1.2.1 through 1.2.3.
Reading this article and completing the exercises should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 3 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use parity in Ken-Ken puzzles.
1.2.2 Counting
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 4 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use counting.
1.2.3 Stacked Cages
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 5 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use the idea of stacked cages.
1.2.4 X-Wing Strategy
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 6 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use the X-wing strategy.
1.2.5 Pair Analysis
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 6 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use the idea of stacked cages.
1.2.6 Parallel and Orthogonal Cages
Note: This topic is covered by the reading assigned below subunit 1.2. Read Section 7 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use parallel and orthogonal cages.
1.2.7 Unique Candidates
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 8 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use the unique candidate rule.
1.2.8 Modular Arithmetic
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 9 of "Using Ken-Ken to Build Reasoning Skills," even though you have not yet studied modular arithmetic. When you get to this part of the course, you will be asked to come back and take another look at this section.
Instructions: Please note that this activity is optional. If you choose to work through this activity, please click on the link above, read the game rules by clicking on the "daily puzzle rules" link, and play a bit.
You should dedicate no more than 1 hour to exploring SET and pick out a few videos to watch on brain teasers. The puzzle will be introduced to you at the beginning of the video. You should pause the video and attempt to solve the puzzle before viewing the solution. Watch the solutions only if you absolutely cannot solve the puzzle; then, go back and reattempt the problem.
You should spend approximately 1 hour on this site, watching a few of these videos and attempting to solve the problems.
Instructions: Please click on the link above and work on the problems on this webpage: liars and truth-tellers puzzles, the Rubik's cube, knots and graphs, and arithmetic and geometry.
Solving these problems should take approximately 2 hours Problems about finding the counterfeit coin among a large group of otherwise genuine coins are quite abundant. Please click on the link above and attempt to solve the problem on this webpage. Solutions appear at the bottom of the webpage. If this type of logical thinking interests you, attempt to find similar problems to solve with an online search.
You should spend approximately 15 minutes attempting to solve this problem.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and read the entire webpage. This text will enable you to see the very close connection between propositional logic and naïve set theory, which you will study in Unit 3. Please note that this lecture covers the topics outlined in sub-subunits 1.5.1 and 1.5.2 as well as any inclusive sub-subunits.
Reading this webpage should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Click on the link above and watch the entire lecture. In particular, focus on the information provided from the 12-minute mark until the 18-minute mark. In this lecture, you will learn which sentences are propositions. Please note that this lecture covers the topics outlined in sub-subunits 1.5.1 and 1.5.2 as well as any inclusive sub-subunits.
Watching this video and pausing to take notes should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Note: This topic is covered by the video lecture assigned below subunit 1.1. In particular, focus on and review the lecture from the 25-minute mark to the 32-minute mark for a discussion on implication and other Boolean connectives.
Instructions: Please click on the link above and read this article, which covers the properties of connectives. While reading, pay special attention to the connection between the Boolean connective and its Venn diagram.
Reading this article and taking notes should take approximately 1 hour.
Instructions: Please click on the links above and read these four sections of Koehler's lectures on logic and set theory. These sections cover the topics outlined below subunit 1.5, including all the sub-subunits.
Reading these sections should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2. Make sure to review the "Logical Operations and Truth Tables" section for an introduction that helps define truth tables.
1.5.2.2 The Boolean Algebra of Propositions
Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2. Make sure to review the section on "Boolean Algebra."
1.5.2.3 Tautologies, Contingencies, and Contradictions
Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2. Make sure to read the definitions of tautology and contradictions (terms highlighted in bold) in the opening paragraphs of the "Logical Operations and Truth Tables" section. Please note that a contingency is simply a proposition that is caught between tautology (at the top) and contradiction (at the bottom). In other words, it is a proposition which is true for some values of its components and false for others. For example "if it rains today, it will snow tomorrow" is a contingency, because it can be true or false depending on the truth values of the two component propositions.
1.5.2.4 Logical Equivalence
Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2. In particular, focus on the text after the heading "Equivalence" toward the end of the "Logical Operations and Truth Tables" section.
Instructions: Please click on the links above and watch these lectures in their entirety to learn about predicates and quantifiers. These videos will also cover the topics outlined in sub-subunit 1.6.1, including 1.6.1.1 and 1.6.1.2.
Watching these lectures and pausing to take notes should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on these webpages above.
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6.
1.6.1.1 Negating Existential and Universal Predicates
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6. Note that the negation of an existentially quantified predicate is a universally quantified one, and vice-versa.
1.6.1.2 The Algebra of Predicates
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6. The main idea here is that predicates can be manipulated in much the same way as numbers, sets, or propositions as we have seen already in the course.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of logical connectives, propositions, negations, quantifiers, truth tables, and counterexamples. When you are done, check your work against those provided in the accompanying solutions file, the Saylor Foundation's "Logic Homework Set Solutions" (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above and read the following sections: "Introduction", "Definition and Theorems", "Disproving Statements", and "Types of Proofs". The types of proofs include Direct Proofs, Proof by Contradiction, Existence Proofs, and Uniqueness Proofs. You may stop the reading here; we will cover the sixth one, Mathematical Induction, later in the course.
Reading these sections should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and read through the examples in the article. The problems are not difficult, but they do serve as clear illustrations of the various aspects of entry-level problem solving.
Reading this article should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and complete this 10-question quiz on logic and related conditionals. Once you choose an answer, a pop up will tell you if you have chosen correctly or incorrectly. You may also click on the drop down menu for an explanation.
Completing this quiz should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above to download the assignment. In order to solidify your problem solving and logical skills, work the eight problems. Then, check your answers against the Saylor Foundation's "Logic Problems Solutions" (PDF).
In this unit, you will explore the ideas of what is called 'naive set theory.' Contrasted with 'axiomatic set theory,' naive set theory assumes that you already have an intuitive understanding of what it means to be a set. You should mainly be concerned with how two or more given sets can be combined to build other sets and how the number of members (i.e. the cardinality) of such sets is related to the cardinality of the given sets.
Instructions: Please click on the links above and read these webpages in their entirety. These texts discuss the basics of set theory. Note that there are three ways to define a set. The third method, recursion, will come up again later in the course, but this is a great time to learn it.
Reading these webpages should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Please click on the link above to download the PDF. Please read pages SF-1 through SF-8 of the file for an introduction to sets, set notation, set properties and proofs, and ordering sets.
Reading this article should take approximately 1 hour to download the PDF. Please read pages SF-9 through SF-11 to learn about subsets of sets. This text also is useful for learning how to prove various properties of sets. If needed, review pages SF1-SF8, which were covered in sub-subunit 2.1.1 above. In particular, please focus on example 9.
Reading these sections should take approximately 30 minutes and read the entire webpage. It is important that you become aware that sets combine under union and intersection in very much the same ways that numbers combine under addition and multiplication. For example, AUB=BUA is a way to say union is commutative in the same way as x + y = y + x says addition is commutative. One difference, however, is that the properties of addition and multiplication are defined as part of the number system (in our development) whereas the properties of sets under the operations we have defined are provable and hence must be proved.
Reading this webpage should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above, scroll down the webpage to week 7, and click on the link for "Boolean Algebra" to download the lecture as a PDF. Please read this entire lecture, paying special attention to the definition of Boolean Algebra and to the isomorphism between the two systems of propositional logic and that of sets. Work the three exercises at the bottom of the PDF and then have a look at the solutions at the end of the document. Note that this reading also covers the topic outlined in sub-subunit 2.2.2.
Reading this lecture and completing these exercises should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read the entire webpage. This brief text will show you how to use characteristic functions to prove properties of sets. However, there are other reasons to learn how to do this. You will see later in the course that functions (not just characteristic functions play a critical role in the theory of cardinality (set size).
Reading this webpage should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and read this webpage, paying special attention to the proof of proposition 3.3.3 at the end of the page. There is a nice proof of this using characteristic functions, which you will be asked to produce later in the course.
Reading this webpage should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and read this webpage, which demonstrates the basic inclusion/exclusion equation outlined in the title of this subunit. The examples on this webpage are especially interesting; pay attention to example 2, which is about playing cards.
Reading this webpage and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of elementary set theory. When you are done, check your work against the answers provided in the accompanying solutions file, the Saylor Foundation's "Elementary Set Theory Homework Set Solutions" (PDF).
This exercise set should take between 2 and 3 hours to complete, depending on your comfort level with the material.
This unit is primarily concerned with the set of natural numbers N = {0, 1, 2, 3, . . .}. The axiomatic approach to N will be postponed until the unit on recursion and mathematical induction. This unit will help you understand the multiplication and additive structure of N. This unit begins with integer representation: place value. This fundamental idea enables you to completely understand the algorithms we learned in elementary school for addition, subtraction, multiplication, and division of multi-digit integers. The beautiful idea in the Fusing Dots paper will enable you to develop a much deeper understanding of the representation of integers and other real numbers. Then, you will learn about the multiplicative building blocks, the prime numbers. The Fundamental Theorem of Arithmetic guarantees that every positive integer greater than 1 is a prime number or can be written as a product of prime numbers in essentially one way. The Division Algorithm enables you to associate with each ordered pair of non-zero integers – a unique pair of integers – the quotient and the remainder. Another important topic is modular arithmetic. This arithmetic comes from an understanding of how remainders combine with one another under the operations of addition and multiplication. Finally, the unit discusses the Euclidean Algorithm, which provides a method for solving certain equations over the integers. Such equations with integer solutions are sometimes called Diophantine Equations.
Instructions: Please click on the link above and read this essay, "Fusing Dots," paying special attention to the exercises at the end. Please note that this reading covers all of the subunits assigned below subunit 3.1. You may find the second half of this reading very difficult. Try to read through Laurie Jarvis' "Understanding Place Value" first (sub-subunit 3.1.12) and then come back to this more challenging paper. You can access the solutions for selected problems here (PDF). Don't worry about understanding all of the details your first time through the reading. Instead, concentrate on the material in the first five sections of the document, and then attempt to generally understand the subsequent sections on Fusing Dots. The supporting details will become more familiar as you work through the various subunits.
Reading this essay should take approximately 2 hours, and completing the exercises should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Note: This topic is covered by the reading assigned below subunit 3.1.In particular, work the problem that involves finding the product of two numbers both given in base 5 notation, without translating to decimal notation.
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.2.1 Using Repeated Subtraction
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.2.2 Using Repeated Multiplication
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.2.3 Translating between Representations
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.2.4 Representing Repeating Base b Numbers as Quotients
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.3 Other Interesting Methods of Representation
3.1.3.1 Base phi Notation
Note: The Base number here is the irrational number phi. This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 6 of "Fusing Dots" and see problem 6 at the end of Section 6.
3.1.3.2 Fibonacci Representation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 6 of "Fusing Dots" and see problem 7 at the end of Section 6.
3.1.3.3 Cantor's Representation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, focus on Section 6 of "Fusing Dots." Problems 12 through 15 at the end of Section 6 all deal with Cantor's representation, also known as factorial notation.
3.1.3.4 Base Negative 4 Notation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 7 of "Fusing Dots." Problem 2 at the end of section 10 is devoted to base negative 4 notation and arithmetic.
Instructions: Please click on the link above to "Prime Numbers" and read the webpage, which includes a good overview of prime numbers and also a list of unsolved problems. Pay special attention to the unsolved problems 1 and 2. Then click on the second link above and read the proof of Euclid.
Reading these webpages should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Please click on the link above and read the entire webpage on Euclid's proof of the infinitude of primes. Be sure you understand why the prime P is not already in the list of primes; if necessary, re-read this text a few times until you have fully grasped this concept.
Reading this article should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Please click on the link above and read this webpage. Take note of the definition of Brun's constant. Also note that this is related to the Intel's famous $475 million recall of Pentium chips. Please also feel free to click on the link to "Enumeration to 1e14 of the twin primes and Brun's constant" link at the end of the webpage to read associated content.
Reading this webpage should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read this brief article to learn a about the Goldbach conjecture. Problems like this are the subject of intense work by mathematicians around the world, and progress is made nearly every year towards solving them.
Reading this article should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the first link above, scroll down to the "Fundamental Theorem of Arithmetic" heading, and read this brief introductory information. Then, click on the second link above and read the entire webpage for information about the Fundamental Theorem of Arithmetic (FTA). Please note that we are going to postpone the proof of FTA until the end of Unit 4. This reading covers the topics outlined in sub-subunits 3.3.1 and 3.3.2.
Reading these articles should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read this entire article on the fundamental theorem of arithmetic. The article may take more time to read than some others. Please note that this reading covers the topics outlined in sub-subunits 3.3.1 and 3.3.2.
Note: This topic is covered by the readings assigned below subunit 3.3. In particular, please focus on Bogomolny's "Euclid's Algorithm" reading and section 2 and 3, "Euclid's Algorithm" and "Alternative Proof", in the Wikipedia article.
3.3.2 Some Applications of FTA
Note: This topic is covered by the readings assigned below subunit 3.3. In particular, please focus on Section 1 "Applications" of the Wikipedia article. Nearly all the proofs of irrationality of the square root of a composite non-square number depend on FTA. Of course, there are also many other applications.
Instructions: Please click on the link above and read this webpage. In particular, focus on the exercise in the reading. Do not be intimidated by the notation in the essay, just read it down to the part on ideals.
Reading this webpage should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the links above and watch these lectures in sequential order. These videos address the concepts outlined in sub-subunits 3.4.1 through 3.4.4. Then, if you chose to work through the Ken-Ken material in subunit 1.2, go to section 9 of the paper "Using Ken-Ken to Build Reasoning Skills" from subunit 1.2, and re-read the section to recall how to use modular arithmetic as a strategy for Ken-Ken puzzles.
Watching these lectures and re-reading this section of "Using Ken-Ken to Build Reasoning Skills" should take approximately 2 hours and 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the first link above and watch this video, which will help you understand the divisibility rules for 3 and 9. Then click on the second link above and watch the entire lecture, which discusses divisibility by 11.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Please click on the link above and try to solve the problem before checking the solution. This problem asks: what is the units' digit of the 2012th Fibonacci number? See if you can work this using your understanding that remainders work perfectly with respect to addition. After you have attempted this problem, review the solution on this webpage.
Completing this assignment should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: By now, the following type of problem should be familiar: what is the units' digit of the expression 7^2012 X 13^2011? See if you can work this using your understanding that remainders work perfectly with respect to multiplication. In other words, if you know the remainder when N is divided by d, then you can find the remainder when N^3 is divided by d.
The solution to this question is mentioned below, but please only check it after you have attempted the problem. After you have completed this problem, click on the link above, and work to solve the problem on this webpage. After you have attempted the problem, click on the link to see the solution.
Solution: The solution to the initial problem mentioned above is that the remainder when N^3 is divided by d is the same as when the r^3 is divided by d, where r is the remainder when N is divided by d.
Completing this assignment should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the first link above and then select the hyperlink to the lecture titled "The Floor or Integer Part Function" to access the PDF. Please read the entire lecture. Then click on the second link, and select the link titled "v.3.0 numberI.pdf" under "1995 Lectures" to access the PDF. Please read the entire lecture.
Reading these lectures and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read this article, paying special attention to Sections 3 and 4, where you will learn about geometry of the divisors of an integer. Complete the problems on the document above, and then check your answers against the Saylor Foundation's "Just the Factors, Ma'am Solutions" (PDF). The topics outlined for subunit 3.6, including sub-subunits 3.6.1 through 3.6.3, are covered by these sections of reading.
Reading this article and taking notes should take about 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages linked above.
Instructions: Please click on the link above and then select the "PDF" link next to "Lecture 6: Number Theory I" to download the file. Please read this lecture, which provides an introduction to decanting (see the Die Hard example on pages 5-7) and the Euclidean algorithm.
Reading this lecture and taking notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.
Instructions: Please click on the link above to download the PDF version of the text and read this paper. This is an easier version of this technique. Solutions to selected problems can be found here (PDF).
Reading this paper should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Note that you have already read this essay in sub-subunit 3.5.1. Please click on the link above, and select the "v.3.0 numberI.pdf" link under "1995 Lectures" to download the PDF. Review the section on "Division Algorithm" again, and then attempt the 3 sample problems in the lecture.
Reviewing this section and attempting the sample problems should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and then click on "Schedule" on the left side of the webpage. Scroll down the webpage to the section "Integer Divisibility," and select the "Linear Diophantine Equations" link to download Lecture 5 as a PDF. Please read this student-friendly version of the lecture, which discusses solving an integer divisibility type of equation. You should focus on solving linear Diophantine equations. In particular, you should be able to find a single solution and then generate all solutions from the one you found.
Reading this lecture should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.
In this unit, you will learn to prove some basic properties of rational numbers. For example, the set of rational numbers is dense in the set of real numbers. That means that strictly between any two real numbers, you can always find a rational number. The distinction between a fraction and a rational number will also be discussed. There is an easy way to tell whether a number given in decimal form is rational: if the digits of the representation regularly repeat in blocks, then the number is rational. If this is the case, you can find a pair of integers whose quotient is the given decimal. The unit discusses the mediant of a pair of rational fractions, and why the mediant does not depend on the values of its components, but instead on the way they are represented.
Instructions: Please click on the link above and read this article. Pay special attention to the five problems on rational numbers at the beginning of the paper. Problem 10 will enable you to appreciate the different between the value of a number and the numeral used to express it. Pay special attention to Simpson's Paradox in the paper. Try the practice problems at the end of the reading. After you have attempted these problems, please check the solutions against the Saylor Foundation's "Fractions Solutions" (PDF).
Please note that this reading covers the topics outlined for subunit 4.1, as well as inclusive sub-subunits 4.1.1 through 4.1.3, and subunit 4.2, as well as sub-subunits 4.2.1 through 4.2.3.
Reading this article and taking notes should take approximately 3 hours. You should also spend approximately 3 hours working on the problems provided in the text.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Note: This topic is covered by the reading assigned below the Unit 4 introduction. In particular, please focus on first 2 paragraphs after the text "Fraction versus rational number," especially where numbers and numerals are in bold font as this will help you understand the relationship between the two.
4.1.2 The Mediant of Two Fractions
Note: This topic is covered by the reading assigned below the Unit 4 introduction. Make sure to work on problem 10 in the essay to help you better understand the mediant of two fractions.
4.1.3 Building New Rational Numbers from Given Ones
Note: This topic is covered by the reading assigned below the Unit 4 introduction. In particular, pay attention to problems 1-3 under "Rational Numbers."
Note: This topic is covered by the reading assigned below the Unit 4 introduction. An interesting property of the rational numbers is that between any two rational numbers we insert another rational number. This property is called density. We say the rational numbers are dense in the real numbers. The same property holds for irrational numbers. Try proving these propositions. Problem 6 in the essay discusses this property.
This topic is also covered in the reading assigned below subunit 4.3.2. In particular, focus on Section 6, "Density of Rational Numbers."
Instructions: Please click on the first link above and watch the video, which shows a proof of the irrationality of the square root of 2. Can you see how to use these ideas to prove that the square root of 3 and of 6 are also irrational?
Next, click on the second link above, and watch the video, which shows a proof that the square root of 3 is irrational. After watching this video, do you think you could prove how the square root of 6 is also irrational?
Watching these videos, pausing to take notes, and answering the questions above should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
Instructions: Please click on the link above to access Professor Tsang's webpage. Select the link to "HW1" to download the PDF. Please read pages 7 through 9, from "Density of Rational Numbers" through "Density of Irrational Numbers." Please note that this reading also covers the topic of Density of Rational Numbers outlined for sub-subunit 4.2.3.
Reading this article, taking notes, and reviewing the proofs several times should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above to access Professor Tsang's webpage. Select the link to "HW1" to download the PDF. Read pages 1-7 of the text. The first 6 pages discuss the field and order axioms for real numbers. The Completeness Axiom on page 6 is what distinguishes the rational numbers from the real numbers – the latter is COMPLETE, while the former is not. This resource covers the topics for sub-subunits 4.4.1 through 4.4.3.
Reading this article and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
In this unit, you will prove propositions about an infinite set of positive integers. Mathematical induction is a technique used to formulate all such proofs. The term recursion refers to a method of defining sequences of numbers, functions, and other objects. The term mathematical induction refers to a method of proving properties of such recursively defined objects.
Instructions: Please click on the links above and watch these brief videos. These videos provide informative discussions as to why the well-ordering principle of the natural numbers implies the principle of mathematical induction.
Watching these videos provides an informative discussion on the principle of mathematical induction and the well-ordering principle of the natural numbers. It specifically addresses the notion of strong mathematical induction.
Watching this video illustrates how the principle of strong mathematical induction can link above and watch the brief video, which illustrates using the principle of strong mathematical induction to links above and read both essays. Notice the similarities between using recursion to define sets and using recursion to define functions. Then answer the four questions at the end of the first essay.
In this type of definition, first a collection of elements to be included initially in the set is specified. These elements can be viewed as the seeds of the set being defined. Next, the rules to be used to generate elements of the set from elements already known to be in the set (initially the seeds) are given. These rules provide a method to construct the set, element by element, starting with the seeds. These rules can also be used to test elements for the membership in the set.
Reading these essays, taking notes, and completing the assignment should take about 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topic of proofs using mathematical induction. When you are done, check your work against those provided in the accompanying solutions file, The Saylor Foundation's "Mathematical Induction Homework Set Solutions" (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.
In this unit, you will learn about binary relations from a set A to a set B. Some of these relations are functions from A to B. Restricting our attention to relations from a set A to the set A, this unit discusses the properties of reflexivity(R), symmetry(S), anti-symmetry(A), and transitivity(T). Relations that satisfy R, S, and T are called equivalence relations, and those satisfying R, A, and T are called partial orderings.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the properties of relations and interrelationships among them, as well as specific examples of relations. When you are done, check your work against the answers provided in the accompanying solutions file, The Saylor Foundation's "Relations Homework Set Solutions" (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above and watch the video. It may be worth spending some time watching this video twice. Note that the lecturer spends some time discussing the definitions of the properties below for sub-subunits 6.1.1 through 6.1.5. The examples he provides exhibit several properties. These are the defining properties of an equivalence relation (see subunit 6.4) and Partial Ordering (see subunit 6.5). Note that this resource covers the topics outlined for sub-subunits 6.1.1 through 6.1.6.
Watching this video twice and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and watch the brief video, which illustrates the notions of relations and functions. This video also provides examples of relations that are functions and some that are not.
Watching this video and pausing to take notes should take approximately 15 minutes.
Note: This topic is covered by the video assigned below subunit 6.2.2. A surjective function is one for which every element in the codomain is mapped to by an element in the domain. For such functions, the codomain and range are equal.
6.2.4 Bijections
Note: This topic is covered by the video assigned below subunit 6.2.2. A bijection is a function that is both one-to-one and onto.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of verifying properties of given functions and equivalence relations, determining if relations are equivalence relations, and commenting on the structure of a relation by using equivalence classes. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation's "Elementary Functions and Equivalence Relations Homework Set – Solutions" (PDF).
This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above and watch the last 10 minutes of this video again. It is especially important that you understand the relationship between an equivalence relation and the partition it induces.
Reviewing this section of the lecture and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
In this unit, you will study cardinality. One startling realization is that not all infinite sets are the same size. In fact, there are many different size infinite sets. This can be made perfectly understandable to you at this stage of the course. In Unit 7.4.3, section (d)iii, you learned about bijections from set A to set B. If two sets A and B have a bijection between them, they are said to be equinumerous. It turns out that the relation equinumerous is an equivalence relation on the collection of all subsets of the real numbers (in fact on any set of sets). The equivalence classes (the cells) of this relation are called cardinalities.
Instructions: Please click on the first link above to watch the video about injections (1 to 1 functions) and surjections (onto functions). Then click on the second link to watch the video, which will show the relationship between injections and functions that have an inverse. Finally, click on the link and watch the last lecture.
Instructions: Please click on the link above and study the proof of Cantor's Theorem. Even though the proof is only one page, this idea is new to you, and therefore is likely to be harder to understand; thus, you should take your time studying this proof carefully.
Studying this proof should take approximately to supplement the written proof of Cantor's Theorem. Then click on the second link and watch the brief video about counting finite sets and Cantor's Diagonalization Theorem.
Watching these videos and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and scroll down to "Handout 8." Select the PDF link to download the file. Read this entire document to learn about countable and uncountable sets. Focus on the several examples of uncountable subsets of R. Please note that this reading also covers topics outlined in subunit 7.2, including sub-subunits 7.2.1 and 7.2.2.
Reading this handout and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of functional properties involving images and inverse images of sets as well as computing images and inverse images of sets. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation's "Functional Properties Homework Set – Solutions" (PDF).
This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of recognizing cardinality properties, determining if sets are countable or uncountable, and determining whether two sets are equivalent. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation's "Cardinality Homework Set – Solutions" (PDF).
This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above and study the proof on this webpage, which shows that rational numbers are countable. Note that information on this topic is also found in the reading assigned below sub-subunit 7.1.3.
Reading this webpage, taking notes, and studying the proof should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read only this webpage; there is no need to click on the "next" or "previous" buttons at this time. Most of these topics have already been covered in the videos for Unit 7, so some of this will be a review.
Reading this webpage should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and select the link to "Course Notes 4.2: Property of Functions" to download the PDF. Please read the paper, paying special attention to examples 2.2.5, 2.2.6, and two functions ax and logax in the paragraph above example 2.7.1. Please note that this resource also covers the topic outlined in sub-subunit 7.3.1.
Reading this paper should take about for an overview of the development of the formulas for the number of permutations and the number of combinations of n objects. For a much more elaborate introduction to counting, click on the second link above, and watch the lecture. This resource covers the topics outlined in sub-subunits 8.1.2 and 8.1.4 below.
Watching these lectures and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read the PDF. Attempt problems 1-20, starting on page 3. Once you have attempted these problems, check your solutions at the Saylor Foundation's "Counting Solutions" (PDF). Please note that this reading and these exercises cover the topics outlined in sub-subunits 8.1.1 through 8.1.4.
Reading this essay and working on these problems should take approximately 5 hours.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the first link above and watch the video for an introduction to the principle. Then, follow this up by clicking on the second link and watching an example of the principle. Notice that the problem is about As, Bs, and Cs, not As, Bs, and Os as the teacher describes at the start.
Watching these videos and taking notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
Instructions: Please click on the link above and watch the brief video. The first video provides some an application of the pigeon-hole principle to divisibility and modular arithmetic. The second video provides some applications of the pigeon-hole principle to operations involving integers.
Watching this video and pausing to take notes should take approximately 30 minutes
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One of a brand new series for A Level students, this book covers all the core content needed for A2 Maths, presented in accessible note form and compiled by top examiners. Individual pages are hole-punched and can easily be removed for insertion into students' own files. "The Collins Revision Notes" series has been specially ... read more
'Achieving in Statistics' is a write-on student workbook that provides a complete revision programme for all seven Achievement Standards in the NCEA Level 3 Statistics and Modelling course. This book can be used as an ongoing study resource for Level 3 statistics students. Once completed, the book becomes a useful set of ... read more
As the Number 1 publisher for GCSE Mathematics, we can offer teachers and students everything they need to succeed in the two-tier AQA specification for 2006. These resources include fully blended Student Books and software to motivate and engage students. Revision and practice books will encourage exam success by consolidati... read more
This maths dictionary is packed with the vocabulary definitions recommended for School years 3 to 6 by the National Numeracy Framework. It contains over 375 entries following the yearly vocabulary recommended for these years and beyond, with headwords, guidewords and a thumb index which follow important dictionary conventions... read more
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The purpose of the this website is to inform parents and students about the Renewed Mathematics Curriculum in Saskatchewan. This site is designed for parents and students of the Greater Saskatoon Catholic School System. If you have any questions or concerns, please contact the GSCS High School Mathematics Teacher-on-Assignment, Shelda Hanlan Stroh, at shanlan-stroh@gscs.sk.ca or (306) 659-7678.
Grade 10 Math Courses
Authored on March 24, 2010 6:46 PM
March 24, 2010 6:46 PM
Students are strongly encouraged by the Ministry and GSCS to take both of the Grade 10 math courses: Foundations and Pre-Calculus 10 & Workplace and Apprenticeship 10.
Rationale:
·Students will benefit as there is rigorous math learning in both courses. This will allow students to develop a deep foundational understanding of mathematics.
·Students should be exposed to the mathematics in all three of the pathways before deciding on a pathway.
·It delays the decision regarding which pathway to take until Grade 11.
·It keeps the students' options open for all three pathways.
About this Entry
This page contains a single entry published on March 24, 2010 6:46 PM.
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College Algebra Concepts and Models
9780618492817
ISBN:
061849281x
Pub Date: 2005 Publisher: Houghton Mifflin College Div
Summary: "College Algebra: Concepts and Models" provides a solid understanding of algebra, using modeling techniques and real-world data applications. The text is effective for students who will continue on in mathematics, as well as for those who will end their mathematics education with college algebra. Instructors may also take advantage of optional discovery and exploration activities that use technology and are integrate...d throughout the text.A brief version of this text, "College Algebra: A Concise Course," provides a shorter version of the text without the introductory review."Make a Decision" features thread through each chapter beginning with the Chapter Opener application, followed by examples and exercises, and ending with the end-of-chapter project. Students are asked to choose which answer fits within the context of a problem, to interpret answers in the context of a problem, to choose an appropriate model for a data set, or to decide whether a current model will continue to be accurate in future years."Chapter Projects" extend applications designed to enhance students understanding of mathematical concepts. Real data is previewed at the beginning of the chapter and then analyzed in detail in the Project at the end of the chapter. Here the student is guided through a set of multi-part exercises using modeling, graphing, and critical thinking skills to analyze the data.Questions involving skills, modeling, writing, critical thinking, problem-solving, applications, and real data sets are included throughout the text. Exercises are presented in a variety of question formats, including free response, true/false, and fill-in the blank.""In the News"" Articles from current mediasources (magazines, newspapers, web sites, etc.) are found in every chapter. Students answer questions that connect the article and the algebra learned in that section. This feature allows students to see the relevancy of what they are learning, and the importance of everyday mathematics."Discussing the Concept" activities end most sections and encourage students to think, reason, and write about algebra. These exercises help synthesize the concepts and methods presented in the section. Instructors can use these problems for individual student work, for collaborative work or for class discussion. In many sections, problems in the exercise sets have been marked with a special icon in the instructor's edition as alternative discussion/collaborative problem."Discovery" activities provide opportunities for the exploration of selected mathematical concepts. Students are encouraged to use techniques such as visualization and modeling to develop their intuitive understanding of theoretical concepts."Eduspace" Houghton Mifflin's online learning tool powered by Blackboard, is a customizable, powerful and interactive platform that provides instructors with text-specific online courses and content and cover are intact. Dust jacket is torn or missing. The book has moderate to heavy wear. Covers have wear; Edges are yellowed and/or dirty; The i [more]
A readable copy. All pages and cover are intact. Dust jacket is torn or missing. The book has moderate to heavy wear. Covers have wear; Edges are yellowed and/or dirty; The inside of the front cover has quite a few smiley faces drawn on it; The corners are worn and bent
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This task requires students to create a matrix from provided data. Students will use the data to respond to several questions that incorporate basic algebra skills such as order of operations and inequalities. This is a great task to use as a pre-assessment for high school students prior to matrix instruction. Student explanations are required. Answer key provided.
Julie Felix - Math Instructional Specialist
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Subject Matter Anxieties
Most studies on achievement anxiety do not differentiate by subject matter. But some people develop anxiety about performance in specific subject areas or with regard to particular skills. They may be comfortable in most academic contexts, but have great difficulty performing in one domain. Two domains that have been studied are mathematics and writing.
Mathematics
Anxious Alma is in good company. Mathematics anxiety, or "mathophobia," is widespread. College students report much more anxiety about mathematics than they do about English, social science (Everson, Tobias, Hartman, & Gourgey, 1993), or even writing (Sapp, Farrell, & Durand, 1995). It is estimated that about one-third of college students suffer from some level of mathematics anxiety (Anton & Klisch, 1995; see Mitchell & Collins, 1991).
A commonly used measure of mathematics anxiety is the Mathematics Anxiety Scale (MAS). The scale includes 10 items and studies of middle school, high school, and college students suggest that it taps the same two dimensions of anxiety often found in more general test anxiety measures—a general sense of worry about mathematics and negative feelings and emotional reactions (Pajares & Urdan, 1996).
Mathematics does not generate as much anxiety in young children as in older children and adults. In Goodlad's (1984) study of over 17,000 young students, mathematics was rated about the same as reading in a list of "liked" subjects (after art and physical education). In the National Assessment of Educational Progress, nine-year-olds ranked mathematics as their best-liked subject; thirteen-year-olds ranked it second best, but in contrast to the younger children, seventeen-year-olds claimed that mathematics was their least liked subject (Carrpenter, Corbitt, Kepner, Lindquist, & Reys, 1981). Significant declines in positive attitudes toward mathematics have also been shown over the adolescent years (Wigfield, Eccles, Mac Iver, Reuman, & Midgley, 1991; Wigfield & Eccles, 1994). Apparently children are not born with mathematics anxiety. Rather, negative attitudes toward mathematics develop over time, especially during adolescence.
Why does mathematics, in particular, cause so much anxiety in older students and adults? One can only speculate. Lazarus (1975) points out that mathematics anxiety has a "...peculiar social acceptability. Persons otherwise proud of their educational attainments shamelessly confess to being 'no good at math'" (p. 281; see also Sapp, 1999).
The way mathematics is usually taught may also explain why mathematics anxiety is common. Lazarus (1975) suggests that the cumulative nature of mathematics curricula is one explanation; if you fail to understand one operation, you are often unable to learn anything taught beyond that operation.
From observations of mathematics and social studies classes, Stodolsky (1985) proposed that mathematics instruction fostered in students the belief that mathematics is something that is learned from an authority, not figured out on one's own. She found that mathematics classes were characterized by (1) a recitation and seatwork pattern of instruction; (2) a reliance on teacher presentation of new concepts or procedures; (3) textbook-centered instruction; (4) textbooks that lacked developmental or instructional material for concept development; (5) a lack of manipulatives; and (6) a lack of social support or small-group work. The instructional format, the types of behavior expected from students, and the materials used were also more similar from day to day in mathematics than in social studies classes. This lack of variety may contribute to anxiety because students who do not do well in the instructional format used in mathematics are not given opportunities to succeed using alternative formats. Later studies by Stodolsky also suggest that mathematics teachers see their subject area as more sequential and static than teachers of other subjects (Stodolsky & Grossman, 1995; see also Wolters & Pintrich, 1998). Sapp (1999) speculates that mathematics teaching often focuses on memorization of procedures, which doesn't prepare students for more conceptual, advanced mathematics. Thus, they feel ill-prepared and become anxious when rote procedures are no longer sufficient.
Stodolsky (1985) also suggests that mathematics is an area in which ability, in the sense of a stable trait, is believed to play a dominant role in performance-either one has the ability or one does not. And if one lacks ability in mathematics, nothing can be done about it. By contrast, people generally believe that performance in other subjects, like reading or social studies, can be improved with practice and effort; they hold an "incremental" theory of ability.
There is consistent evidence that females suffer more from mathematics anxiety than do males (Hembree, 1990; Pajares & Urdan, 1996; Randhawa, 1994; Wigfield & Meece, 1988). Some researchers have proposed that mathematics anxiety contributes to observed gender differences in mathematics achievement and course enrollment, but the one study that actually assessed anxiety and enrollment plans found no relationship (Meece, Wigfield, & Eccles, 1990).
There is little agreement on the reasons for such gender differences. Ability differences, socialization differences, differences in the level of self-confidence, and the number of mathematics courses taken have all been proposed as explanations. Whatever the reasons for the frequency and intensity of mathematics anxiety, particularly among females, it is a problem that warrants special attention by educational researchers and practitioners.
The good news is that interventions to reduce math anxiety have been successful. Sgoutas-Emch and Johnson (1998) found that writing in a journal about frustrations and feelings reduced college students' anxiety in a statistics course.
Writing
Perhaps everyone, at one time in their lives, experiences a certain amount of panic facing a blank piece of paper or computer screen, especially if the due date for a written product—a paper for a class or a report for work—is close at hand. "Writer's block" is so debilitating for some that they avoid courses and professions that require writing. (See Daly & Miller, 1975b; Daly, Vangelisti, & Witte, 1988; Rose, 1985; Selfe, 1985.)
Although psychoanalytic explanations have been suggested (Barwick, 1995; Grundy, 1993), the few studies that have been done suggest that writing anxiety reflects some of the same dynamics that explain general achievement anxiety. Writing anxiety, like general achievement anxiety, is associated with relatively low expectations for success as well as lower writing quality (Daly, 1985; Pajares & Valiante, 1997). Rose's (1985) research on writer's block makes it very clear that the causes are usually multifaceted, and that although they may have their roots in early familial experiences, later and current experiences in writing contexts are also important.
Researchers have developed a measure of writing anxiety (Daly & Miller, 1975a), which has been shown to be more strongly associated with writing performance than a more general measure of achievement anxiety (Richmond & Dickson-Markman, 1985). Studies using the measure have found some gender differences, with females showing somewhat less writing anxiety than males. People high in writing anxiety were also high on reading anxiety and anxiety about public speaking and interpersonal communication, but relatively low on math anxiety (Daly, 1985).
Research has also examined associations between teachers' feelings about writing and their teaching strategies. Studies have found, for example, that highly apprehensive female teachers assign fewer writing assignments and are more likely to be concerned with issues of form and usage and less likely to emphasize personal or creative expression and effort than less apprehensive teachers (see Daly, 1985; Daly et al., 1988). Associations between teachers' own anxiety about writing and their teaching methods were strongest in upper elementary school, when many important writing skills are supposed to be taught.
The instructional context can exacerbate concerns about competencies that feed anxiety, and they contribute to trait anxiety both over time and collectively. One study found that high school students who were relatively high in writing anxiety reported having experienced more criticism for their writing and less encouragement and support, and they reported seeking help for writing problems less than students low on writing anxiety (Daly, 1985).
Daly (1985) proposes that writing anxiety will be greatest under the following circumstances:
evaluation is salient
the task is ambiguous
the writer feels conspicuous
task difficulty is perceived to be high
the writer feels lacking in prior experience relevant to the task
the task is personally salient
the setting or task is novel
the writer perceives the audience as uninterested but evaluative
Teachers may be able to reduce writing anxiety by minimizing students' concerns about evaluation, making assignments and criteria for grading clear, and making sure that students have the prior experience and familiarity they need to complete the writing task. Writing tasks, like all tasks, should be challenging but not so difficult or different from what students have experienced in the past as to provoke a sense of incompetence or low expectations for success. A genuine and supportive audience (e.g., classmates, parents) might also help.
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Stop searching through textbooks, old worksheets, and question databases. Infinite Pre-Algebra helps you create questions with the characteristics you want them to have. It enables you to automatically space...
Basic Algebra Shape-Up is interactive software that covers creating formulas; using ratios, proportions, and scale; working with integers, simple and multi-step equations. Students are able to track their...
Answers.com is a revolutionary Answer Engine: a one-click reference library on demand. With 1-Click Answers software, just Alt-click on any word in any Windows application (e-mail, browser, Office, and so...
Problem Solved! 4.6 is a comprehensive but easy to use help desk program which includes both a windows based application and a mini-http server that you have running in less than an hour. The web interfaceThis bilingual problem-solving mathematics software allows you to work through 5018 Algebra equations with guided solutions, and encourages to learn through in-depth understanding of each solution step and...,...
This script defines the Matrix class, an implementation of a linear Algebra matrix. Arithmetic operations, trace, determinant, and minors are defined for it. This is a lightweight alternative to a numerical...
Passguide Vmware VCP-510 testing engine simulates a real VCP-510 exam environment. Questions & Answers are compiled by a group of Senior IT Professionals. Questions are updated frequently - you get a free...
Algebra Helper shows how signs change when moving a quantity from one side of an equation to the other. Algebra Helper is implemented in JavaScript. Currently works in Firefox and Internet Explorer 8....
Boolean Algebra assistant program
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Interactive College Algebra course designed to ensure engaging, self-paced, and self-controlled e-learning process and help students to excel in their classes. Java- and web-based math course includes... Answers...
Mavscript allows the user to do calculations in a text document. Plain text and OpenOffice Writer files (odt,sxw) are supported. The calculation is done by the Algebra system Yacas or by the Java interpreter...
FAQ Search allows you to place a searchable and easy-to-update Frequently Asked Questions database on your site in minutes. One script file, and a single database text file can be uploaded in a few minutes...
MathProf is an easy to use mathematics program within approximately 180 subroutines. MathProf can display mathematical correlations in a very clear and simple way. The program covers the areas Analysis,...
Calculator is a minimalist, easy to use Windows calculator that takes convenient advantage of the number key pad in your keyboard. Calculator can compute any Algebra expression instantly.
Calculator runs...
It is one of the most powerful math tools there is. It gathers between the simplicity and ease of use of a simple calculator and the ability to solve complex math procedures.
Here is a brief description of...
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The diagram below illustrates the sequence of developmental courses that lead to these credit math courses. Your starting point in this sequence is determined by predefined criteria or placement testing. The blue boxes illustrate the Algebra Pathway and the green boxes illustrate the Statistics Pathway.
Credit Math Course Descriptions:
MATH 1314 (College Algebra): The course is designed to prepare students for higher level math courses such as calculus and business math and to teach the math skills required for related science courses such as chemistry and physics.
MATH 1332 (Contemporary Math): The course uses multiple approaches such as physical, symbolic, graphical, and verbal to solve financial and social application problems.
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How to convert a geometric series so that exponent matches index of sum? Fair enough. But here in Britain for example, we don't have a specific course that prepares us for calculus. (Indeed, we generally start studying calculus at a younger age, and the age is even younger in parts of Europe.) I suppose we don't even have a rigorously delineation of fields of mathematics when we study, so it wouldn't make much sense to us anyway.
May
27
comment
How to convert a geometric series so that exponent matches index of sum? Out of curiosity, why is this tag "precalculus" in use? As far as I know it's a strange word only used in the USA, and in any case is an umbrella term for some very disparate subfields of basic mathematics. Most Europeans (and I believe I speak for Asians too) would have no clue what it means.
Send a File to the Recycle Bin You don't know what DLLs will be available on the system though, in the future. If anything is likely to be removed, it's a DLL like this. Though MS are obsessed with backwards-compatibility, so perhaps none will ever get removed. Yay for bloat?
Can a monad be a comonad? @AndrewC: No need to be combative mate; I'm not saying this question should be closed. I was just offering a helpful tip to the questyion asker. I actually agree with you... category theory can be a highly abstract and theoretical tool in mathematics, or it can be a rougher conceptual underpinning of functional programming, or anywhere inbetween.
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Head First Geometry Geometry Book Description
Having trouble understanding geometry? Do vectors, parallelograms and number-filled proofs just make your head spin? Full of engaging stories and practical, real-world explanations, Head First Geometry will help you learn everything from the quadrilateral equation to exploring three-dimensional shapes to finding similar triangles. Along the way, you'll go beyond solving hundreds of repetitive problems, and actually use what you learn to make real-life decisions. You've found a really cool design to etch on the back of your new iPhone, but will it actually fit on there? Learn how to put Geometry to work for you, and nail your class exams along the way. Why waste your time struggling with new concepts? By using the latest research in cognitive science and learning theory to craft a multi-sensory learning experience, Head First Geometry uses a visually rich format designed for the way your brain works, not a text-heavy approach that puts you to sleep.
About the Author :
Lindsey Fallow has contributed to Head First Geometry as an author.
s
Popular Searches
The book Head First Geometry by Lindsey Fallow, Dawn Griffiths
(author) is published or distributed by O'Reilly Media [0596801289, 9780596801281].
This particular edition was published on or around 2009-10-31 date.
Head First Geometry
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This would make a great project for polynomials. Students can either take or find pictures on the internet, overlay a coordinate grid and the curve and then write the function that would produce that curve. Look at more examples here.
In middle school, there's little need for 3D graphing, but sometimes it's good to show students where they're going and not simply where they are. It also has 2D capability which would be more useful in regards to graphing linear equations or quadratic equations for Algebra.
It took less than a minute to install, and if you're familiar with graphing calculators at all, it's fairly self explanatory. For the Algebra 1 classroom, the basic functions you would need to know are:
1) Input your equation(s) in the "y=" on the top left.
2) Click on the coordinate plane icon at the bottom left if you want to change the default 5×5 grid to 10×10.
3) The table icon near the top middle will hide/show data.
It doesn't do too much more than the TI software your school might have except that it's more colorful and requires no set-up. But one school I taught at had very little technology resources at all – I wish I had known about this back then.
There are a few bugs, nothing that will affect your computer and nothing really important. For instance, the "example" button will show you various graphs. Each time you click this example button, a new graph appears. There's only 10 or so that it cycles through but if you switch from 2D examples to 3D, then it'll just stop graphing anything even though the data still appears to change. Well, like I said, this isn't important.
Also, with this free version your capabilities are limited. Many of the functions are disabled such as the ability to save, etc. The most annoying part about that is that if you accidentally click on an icon that has been disabled, it will ask you if you want to upgrade (pay about $50 for the real deal). Which I understand since this is a free version, and they'd prefer to sell their product. But I think this free version is all that you would need for a middle school class, though maybe it would be worth purchasing for Calculus or Statistics (they have an advanced Stats calculator).
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Advanced Placement Program* Summer Institute
AP* Calculus AB, July 22-25, 2013
Lead Consultant: Candace Smalley
What to Bring
A graphing calculator
Experienced teachers should bring an AP* level activity to share with participants
Course Description
This session is specifically designed to help interested teachers build a successful AP* Calculus AB course. The week will include an analysis of the current curriculum, including an examination and discussion of various teaching strategies that reflect the current philosophy and goals of the course. Included will be an overview of the AP* program; suggestions for pacing and sequencing of concepts; a study of numerous AP* level problems; activities with graphing calculators (including CAS systems); a review of the AP* Exam including format, scoring standards and student responses; a discussion of the grading process from the perspective of an AP* Table Leader; and an overview of available resources and materials for AP* teachers.
About Candace Smalley
Candace Smalley currently teaches mathematics at Trinity Valley School in Fort Worth, TX after retiring from her teaching position in Oklahoma where she taught for twenty-two years. In Oklahoma, she taught the AP* Calculus AB course since 1995 and the AP* Calculus BC course since its inception in the district in 2000. At Trinity Valley School she currently teaches Calculus, AP* Calculus AB and an Advanced Calculus/AP* Calculus BC class.
Candace has served as a College Board Consultant since 1998 and is a table leader for the AP* Calculus exams. She has served on the College Board's Southwest Region Advisory Council and the Southwest Region Conference Planning Committee. She has been a presenter at many AP* Conferences including a recent workshop in Hong Kong, and lead instructor at numerous AP* summer institutes. Candace is a recipient of the College Board's Advanced Placement Special Recognition Award and also received recognition for her work with the AP* program as a Siemens Award for Advanced Placement winner.
2012 Participant Testimonials
"Wonderful workshop. Very informative. Great presenter."
"This course is excellent. The teacher did an outstanding job conducting this course. She did a great job integrating all the materials and using different resources."
"I especially loved learning how she teaches students and what she uses as her hooks!!"
* Advanced Placement Program and AP are registered trademarks of the College Board and have been used with permission.
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Algebra and Trigonometry Algebra and Trigonometry. This text presents the traditional content of the entire Precalculus series of courses in a manner that answers the age-old question of When will I ever use this? Highlighting truly relevant applications, this text presents the material in an easy to teach from/easy to learn from approach. This book presents the traditional content of Precalculus in a manner that answers the age-old question of "When will I ever use this?" Highlighting truly relevant applications, this book ... MOREpresents the material in an easy to teach from/easy to learn from approach.KEY TOPICS Chapter topics include equations, inequalities, and mathematical models; functions and graphs; polynomial and rational functions; exponential and logarithmic functions; trigonometric functions; analytic trigonometry; systems of equations and inequalities; conic sections and analytic geometry; and sequences, induction, and probability. For individuals studying Precalculus.
Quadratic Functions. Polynomial Functions and Their Graphs. Dividing Polynomials: Remainder and Factor Theorems. Zeros of Polynomial Functions. More on Zeros of Polynomial Functions. Rational Functions and Their Graphs. Modeling Using Variation.
The Law of Sines. The Law of Cosines. Polar Coordinates. Graphs of Polar Equations. Complex Numbers in Polar Form; DeMoivre's Theorem. Vectors. The Dot Product.
8. Systems of Equations and Inequalities.
Systems of Linear Equations in Two Variables. Systems of Linear Equations in Three Variables. Partial Fractions. Systems of Nonlinear Equations in Two Variables. Systems of Inequalities. Linear Programming.
9. Matrices and Determinants.
Matrix Solutions to Linear Systems. Inconsistent and Dependent Systems and Their Applications. Matrix Operations and Their Applications. Multiplicative Inverses of Matrices and Matrix Equations. Determinants and Cramer's Rule.
10. Conic Sections and Analytic Geometry.
The Ellipse. The Hyperbola. The Parabola. Rotation of Axes. Parametric Equations. Conic Sections in Polar Coordinates.
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Algebra
1
Chapter 1 Assignment Calendar
Note: All
problems should be done without a calculator unless the number is in bold.
Day 1_______Test over Prerequisite material
Section 1.1
– Variables in Algebra
Assignment: page
6 # 2, 3, 6-12 (even), 13 –16,
19 – 30, 32, 35 – 41, 45, 47, 50 – 68
Day 2 – Section 1.2 – Exponents and Powers
Homework: Section 1-2: pp. 12-14(#4-7, 9-13, 24, 25, 55, 56, 61, 64,
72-78 even)
Section 1.3 – Order of Operations
Homework: Section: 1-3: pp19-22 (#4-11, 24-30,
38, 39, 42,
52, 56a, 56b, 72)and
Prepare for quiz next class!!! Quiz on pg.
22 is a good review
but not required.
Day 3 – Section 1.4 –Equations
and Inequalities
Section 1.5 A Problem solving Plan Using Models
In Class:Quiz
Sections 1.1 – 1.3
Assignment: pp 27-30 # 1 – 6, 14-24 even, 38-48 even, 56-59, 70-72)
pp 35-38 # 1
– 3, 5 – 12, 39 – 45, 67 – 72
Day 4 –Section 1.6 –Tables
and Graphs
Assignment: pp43-45 # 4 – 10, 12 – 17
Section 1.7 – An Introduction to Functions
Assignment:
pp 49-51 # 13
– 16, 19-24, 49, 50, 56, 57
EXTRA CREDIT OPPORTUNITY: Use one of the graphs
we presented about in class today and make up a new story relating 2 factors,
for example time and money, time and heart rate, etc. Be prepared to present it
to
Day 5 - Chapter Review
class work: pre-test and 1.7 worksheet
Homework: pp
57 # 1 – 29
and finish 1.7 w/s
Have chapter 1 notes ready to hand in.
Day 6 –Test on Chapter 1
2.1 The Real Number Line
Quotations
by Aristotle
There are things which seem
incredible to most men who have not studied mathematics.
Quoted in S Gudder, A Mathematical Journey
A nose which varies from the
ideal of straightness to a hook or snub may still be of good shape and
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Christina Zarb
Applying mathematics
One of the questions every mathematics teacher is regularly asked is "When are we ever going to use these things?" This is probably the hardest question to answer in a satisfactory manner! We talk about the applications of mathematics in physics, engineering and computer science.
It is today becoming more important for students to attain a higher level of numerical literacy
- Christina Zarb
Since I deal with many IT students, my favourite answer is "Google! The Page Rank algorithm, which is used by the Google search engine, is a highly successful application of graph theory and matrices!" That catches the interest of a few. But what about the students who are not interested in pursuing one of these areas? How can we make mathematics more relevant to those who want careers in finance, economics and the social sciences?
Of course we can, many should be thinking. Experts in these fields know that mathematics and quantitative analysis are of fundamental importance. But are our students, especially those who study mathematics at post-secondary level, being given the opportunity to hear about these applications of mathematics?
Mathematics is a compulsory subject throughout primary and secondary education – it is considered, and rightly so, a core subject throughout these highly important formation years, and a pass at Ordinary level is fundamental in order to continue one's studies at post-secondary and tertiary level.
After secondary school, students have to choose which combination of Advanced and Intermediate level subjects to pursue, keeping in mind the University degree they would like to read for. Of course these include Mathematics in two different forms, the classic Pure Mathematics and Applied Mathematics, offered both at Intermediate and Advanced level.
Pure Mathematics provides a basis of the main important areas in mathematics, including calculus, vector and matrix analysis, numerical methods and complex numbers. It is a necessary choice for students who wish to follow a career in Architecture, Engineering, IT, Accounting or Physics.
Applied Mathematics is essentially a course in Mechanics and is highly recommended, mainly at Intermediate level, for students interested in Engineering and Physics.
And this is where I feel there is a lacuna in Mathematics education at this level. The syllabi, both at Advanced and Intermediate level, in fact specify that this is a course in mechanics. But what about all the other forms of Applied Mathematics?
The qualification of Applied Mathematics with the description "(Mechanics)" rightly seems to imply that there are other forms of Applied Mathematics: Discrete Mathematics, highly relevant to Computer Science and even to Operations Research; Financial Mathematics, for anyone interested in Finance and Economics; Quantitative Analysis and Statistics, of primary importance to any Social Science such as Psychology, Sociology, Management. But at present there is still no provision for these forms of Applied Mathematics at this level.
If we look at the current British educational system, which is very often the main model which Malta follows, these extremely important topics have been provided for at this level for many years. The British Advanced (A) level and Advanced Supplementary (AS) level system works on the modular concept. Students can choose from a number of modules, including Pure Mathematics, Statistics, Mechanics and Decision Mathematics, and these serve as building blocks for the qualification required – three modules to achieve an AS level certificate, and six modules to complete an A-level certificate in the subject.
There are compulsory core Pure Mathematics modules, and then students are free to "mix and match" units which give them a flavour of the various forms of Applied Mathematics – of course, one can always choose six modules in Pure Mathematics, which leads to the traditional Pure Mathematics A-level.
Other options include A/AS-level Mathematics, which can include Statistics, Mechanics and Decision Mathematics modules, as well as A/AS-level Statistics. Statistics and Quantitative Analysis are central to many areas, including physical sciences, life sciences, social sciences and financial subjects and, in my experience, many students at tertiary level struggle with these subjects, particularly especially since they would often have stopped studying any form of numeracy at age 16.
Looking back at the local examinations, the provision of Statistics at Intermediate level (and possibly A-level) as part of the Matriculation certificate is surprisingly missing, particularly since Statistics and Operations Research are now available at Bachelor and Master of Science level at the University.
Decision Mathematics modules include areas of operations research and graph theory and its applications, and are of particular interest to computer science students, as well as to anyone interested in logistics and operations management.
Going back to the British system, there is another more recent AS-level option called the Use of Mathematics, specifically intended for "non-mathematicians", and the emphasis here is on modelling and communicating real-life situations using mathematics.
Perhaps similar options at Intermediate Level will lead to more students taking up some form of Mathematics at post-secondary level, facilitating their tertiary level studies and making mathematics more relevant in today's competitive world.
There has been much debate on post-secondary Mathematics education in the UK recently. Studies have shown that more students have been taking A-level Mathematics in some form. Yet studies commissioned by the Education Ministry, among others, have reported that, compared to other countries, the proportion of students studying mathematics beyond the age of 16 is much lower than that in other OECD countries.
I quote from a report published by ACME (Advisory Committee on Mathematics Education "A major problem in England, together with Wales and Northern Ireland, is the two-year gap for most young people between the end of GCSE and the start of university or employment, during which the large majority do no mathematics. This is the most striking and obvious difference between the mathematics provision here and that in other comparable countries. It is not just a case of students missing out on two years of learning mathematics, serious though that is, but of their arriving at the next stage of their lives having forgotten much of what they did know."
This is comparable to the situation in Malta, where the education system is quite similar, and many students who do not feel competent enough in Mathematics do not choose any form of Mathematics as part of their Matriculation certificate.
According to the Matriculation Certificate Report for 2011, 25 per cent of all students chose Pure/Applied Mathematics at A-level, and 24 per cent chose Pure/Applied Mathematics at Intermediate level. It would be interesting to carry out a study which would indicate which University courses these students pursued, as well as reviewing the course choices by students who did not choose any form of mathematics as part of the Matriculation certificate, in order to establish whether they would have benefited had they chosen a numerical subject.
In the education sector, a lot of emphasis is currently being given to the National Curriculum Framework, which covers the compulsory education years. An entire document is dedicated to science education, entitled "A Vision for Science Education in Malta".
The main targets of the reform in this area include an increase in the number of students who take up science courses at post-secondary and tertiary level, as well as an improvement in Malta's performance in international education surveys in the science area, specifically mentioning the TIMSS (Trends in International Mathematics and Science Study) and PISA (Programme for International Student Assessment) urveys in which Malta has recently started participating.
In the PISA 2009 results, which measure mathematical literacy for 15-year-olds in 74 countries, Malta classified in the 40th position, with an average score below the OECD average, and below that of most EU countries.
This does not bode very well for science education in general, because it is widely accepted that strong mathematical literacy is essential for studying science at post-secondary and tertiary level.
For these reforms to bear the desired fruit, I believe one must also view the wider picture of post-secondary education, particularly, the years between the end of compulsory education and the start of a university degree, and have a more cohesive and seamless system up to the age of 18/19. And it should be a system which takes into consideration the teaching of mathematics within the wider context of the teaching of science, something which "A Vision for Science Education in Malta" fails to do.
I hope this article will contribute to a debate about mathematics education in the post-secondary phase. Mathematics in its pure form is beautiful to some, a night-mare to others. It is undeniable that, in today's world, numerical literacy of some form is very important – perhaps opening up access to new professions, without ever killing the classical ones which are already available.
For many courses at tertiary level it is today becoming more important for students to attain a higher level of numerical literacy before they move on to become part of the labour force. With the government's vision of making Malta a centre of excellence in IT and Finance, it is crucial that our future graduates have a deep understanding of mathematics, and the capabilities to apply it to these and similar areas.
Ms Zarb lectures in Mathematics and Statistics at St Martins Institute of IT opening sentence of this article refers to a question which probably exposes a recurrent failure in the teaching or indeed, the exposition of the subject matter.
Those professionals who teach mathematics may wish to seek alternative ways in conveying the subject to thier pupils. A comparison using various self devised techniques say over a time span of 5 years, would probably result in an enhanced appeal of the subject to other potential students, who would "normally" fall behind others !!
Fully agree with your suggestions... hope authorities take note..especially the Faculty of Education and the Faculty of Science and Matsec Board.
Perhaps the decision makers ought to try teaching for once to really understand the situation.... Most are just cut off from reality. Reforms stopped at secondary education.Post secondary..quo vadis? Intermediates are just a joke -most students takes serious interest in the subjects and just study for these exams at the very end..
When are we ever to use Mathematics?
The interesting part about that question is that whether we use them or not, mathematics are the operators that describe the laws of nature! Whether we learn to use them or not, well, there they are being applied continuously for as long as the universe exists!
Mathematicians arelike linguists, who invented a set of noises to describe to other people what goes in their mind. Mathematics are operators invented by man which describe the actions of our universe.
If a child plays with a ball on the surface of the earth , the moon or in space, we can predict the trajectory it follows by applying mathematics.
If mother cooks a meal, the amount of the various components used to ensure that the recipee is repeatable, we have to use mathematics.
The clothes we wear and the curtains on the wall and the amount of paint we need to paint a room, well it is all described by mathematical operators. Here is a surprise for many people. If I make a model of a boat and then I make a bigger boat which is 4 times as big, how much more paint would I need and how much more material would I neer. here is the surprise. If the boat is four times as big then assuming the paint is of equal thickness as that used in the model then then amount of paint will be 16 times as much as paint covers AREA. Now the material covers VOLUME and so makeing a boat four times as big as the model will need 64 times as much material!! Many people are surprised when then buy a house twice as big and then the costs run to much more than twice. Building a house uses area for some needs and volume for other uses and the prices is all calculated thrugh mathematics.
If one is buying a team for football or volley ball, it would be stupid to ask " What is their average height?" as one might get a surprise for out of 10 players five could be 9 foot tall and the other five could be one foot tall.
No mathematics overcome all that bu using other functions so that the right questions are asked and inthis case it would be more intelligent to ask," What is the MEAN DEVIATION of the players in the team. SO IF YOU KNOW MATHEMATICS NO ONE WILL FOOL YOU WHEN BUYING POWERSTATIONS, AIRPORTS AND SHIPS AND AIRPLANES AND CLOTHES AND ALL THAT WE BUY INCLUDING WASHING MACHINES AND TELELVISION.
Enough said as if I carry on I will not stop.
One thing I ask of mathematicains, please introduce mathematics so that young peole can understand, as the manner in which this article was written , IT WILL SCARE YOUNG PEOPLE AWAY FORM SCIENCE AND MATHEMATICS. Take it easier when teaching mathematics as the concepts of operators to young people needs careful and cling introductios, Please do take an easier introduction , and depict al the mathemtics in games and sport and such like situations including domestic circumstances including a wald in the countryside and just a strole around the markets at Vittoriosa and Marsaxlokk. The maths in a home are infinite and at Paceville it will surprise people what they will find if they act like "Johnny head in air "and if only look around mathematics will sow itself provided the introduction is reasonable and slow. If only one could recognise that behing every action there is a law that governs it, then that is a good manner to recognise mathematics as ater all maths are like the spirit and the soul of every action. We only see the action but the spirit and sould are there behind it .
Prosit for this article but mathematics must be introduced in a somewhat easier vein.
Nice of you to mention the Square Cube Law :-)) Like a plane's wing area (square function) and cabin volume (cubic fumction) !! They just don't double up !!
"Mathephobia" is recurrent problem and I think it's related to how the subject is conveyed. I knew a maths teacher once who would have made the perfect extra in the opening scenes of Stanley Cubric's "Full Metal Jacket" !!
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Quick Links
Resources
Math Department
General Math Classes
ALGEBRA I
The purpose of this course is to provide students with the foundation for more advanced mathematics courses and to develop the skills needed to solve mathematical problems. Course content includes sets, variables, real number systems, equations and inequalities, relations and foundations, graphs, rational and irrational numbers, and radicals.
ALGEBRA II
The purpose of this course is to providestudents with a foundation for applying advanced skills to other mathematical and scientific fields. Course content includes linear and quadratic equations, factoring of polynomials, graphing systems of equations, and rational/irrational functions.
GEOMETRY
This course emphasizes critical thinking involving the discovery of relationships and their proofs, and skills in applying the deductive method to mathematical situations. Course content includes logic and reasoning, the study of Euclidean geometry of lines, planes, angles, triangles, similarity and congruence.
ANALYTIC GEOMETRY
Students will use graphing technology to find approximate solutions for polynomial equations. Write the equations of conic section s in standard form and general form, in order to identify the conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.)
TRIGONOMETRY
The purpose of this course is to teach students to make connections between right triangle ratios, trigonometric functions, and circular functions.
MATH FOR COLLEGE READINESS
The purpose of this course is to strengthen the skill level of high school seniors who have completed Algebra I, II, and Geometry and who wish to pursue credit generating mathematics courses at college level. Course content includes: Functions and Relations: Polynomials; Rational Expressions and Equations; Radical Expressions and Equations; Quadratic equations and Strategies for College Readiness. Exit requirements must pass with a "C" or better. Pass CPT: above 86, SAT: above 520, ACT: above 22. If the students meets the above exit requirements, the student will not have to take any prerequisite classes and must enroll in College Algebra within two years of completion of this course.
PRE-CALCULUS
The purpose of this course is to enable students to develop concepts and skills in advanced algebra, analytic geometry, and trigonometry. The content should include: trigonometric functions and their inverses, trigonometric identities and equations, vectors and parametric equations, structure and properties of the complex number system, polar coordinate system, sequences and series, concept of limits, conic sections, polynomial, rational, exponential, and logarithmic functions, and matrix algebra.
College Level Math Classes
AP CALCULUS AB
AP Calculus AB is designed to familiarize the student with the basic concepts of introductory calculus. It is equivalent to a first semester course in college. Course content includes functions, limits and continuity, derivatives and integrals. There is a strong emphasis on application through topics such as related rates, optimization, volume of rotation work and moments.
AP STATISTICS
The purpose of this course in statistics is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: Explore Data: Describing patterns and departures from patterns, Sampling and Experimentation: Planning and conducting a study, Anticipating Patterns: Exploring random phenomena using probability and simulation Statistical Inference: Estimating population parameters and testing hypotheses.
Math Electives
INTENSIVE MATHEMATICS
The purpose of this course is to enable students to develop mathematics skills and concepts through remedial instruction and practice. The content includes, mathematics skills that has been identified by screening and individual diagnosis of each student's need for remedial instruction and specified in his/her Progress Monitoring Plan, critical thinking, problem solving, and test-taking skills and strategies. This course counts as an elective.
MATH FOR COLLEGE SUCCESS
The purpose of this course is to prepare students for entry level College Mathematics. Major topics include properties of integers and rational numbers, integer exponents, simple linear equations and inequalities, operations on polynomials including beginning techniques of factoring, introduction to graphing, and introduction to operations on rational expressions. The content should include: using signed numbers; simplifying algebraic expressions; solving algebraic equations; simplifying exponents and polynomials; factoring polynomials; graphing linear equations; simplifying, multiplying, and dividing rational expressions; simplifying and performing operations with radicals. Exit requirements: Obtain grade of "C" or better and pass the state exit exam at the identified "cut-score". If the student meets the above exit requirements, the student will not have to retake the CPT or have to take any non-college credit classes within two years of completion of this course.
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Transition Mathematics
Main goal: The main goal of Transition Mathematics is to act as a stepping-stone between the processes learned in Pre-Transition Mathematics or Everyday Mathematics 6 to the material presented in UCSMP Algebra and UCSMP Geometry. Transition Mathematics incorporates applied arithmetic, algebra, and geometry; and connects all these areas to measurement, probability, and statistics.
Main theme I: Arithmetic skills and concepts are reinforced by continuous instruction in the uses of the four basic operations of addition, subtraction, multiplication, and division. Basic skills and number sense practice are reinforced by applications and the conversions among decimals, fractions, and percents, with both positive and negative numbers.
Picturing multiplication by 2.5 and by 0.8
Main theme II: The algebra in Transition Mathematics begins with the uses of variables in formulas, as pattern-generalizers, and as unknowns in solving problems. Graphing lines in the coordinate plane and the solving of linear equations and inequalities are developed.
Main theme III: The geometry in Transition Mathematics includes the use of transformations to demonstrate congruence, similarity, symmetry, and tessellations. Length, perimeter, area, and volume are studied as general concepts and with specific attention to common two- and three-dimensional figures. Drawing and constructions with and without the use of technology are both strongly encouraged throughout the text.
Comparisons between this and earlier editions: The reality orientation of the material and the overall approach of this groundbreaking book remain. Some of the content in the first and second editions of Transition Mathematics has been moved to Pre-Transition Mathematics due to (1) the existence of Everyday Mathematics and the general increase in the performance of students coming into middle school, (2) increased expectations for the performance of all students in both middle and high schools and the concomitant increased levels of testing, and (3) recommendations for more algebra and geometry in middle school courses preceding year-long courses in algebra and geometry. Calculators with graphing and list features are introduced early as pattern-fitting and problem-solving tools. Spreadsheets and dynamic geometry systems are found in activities throughout the materials. Students are engaged and learning is reinforced with the use of games.
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Algebra 1
Instructor's Annotated Edition
ISBN/ISBN10
978-0-495-38988-0
0-495-38988-9
Order#
1090340
Price
$202.75
Quantity
This special version of the complete student text contains a Resource Integration Guide to using the ancillary teaching and learning resources with each chapter of the text, as well as answers printed next to all respective exercises. Graphs, tables, and other answers appear in a special answer section in the back of the text.
Titles marked with asterisk (*) indicate product is restricted from sale to individuals and may only be purchased by a registered institution. Go here if you are not already logged in or need to register.
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9780314069Thomson Advantage Books: Algebra and Trigonometry
In this new ADVANTAGE SERIES version of David Cohen's ALGEBRA AND TRIGONOMETRY, Fourth Edition, Cohen continues to use the right triangle approach to college algebra. A graphical perspective, with graphs and coordinates developed in Chapter 2, gives students a visual understanding of concepts. The text may be used with any graphing utility, or with none at all, with equal ease. Modeling provides students with real-world connections to the problems. Some exercises use real data from the fields of biology, demographics, economics, and ecology. The author is known for his clear writing style and numerous quality exercises and applications. As part of the ADVANTAGE SERIES, this new version will offer all the quality content you've come to expect from Cohen sold to your students at a significantly lower price
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Brief Calculus And Its Applications - 8th edition
Summary: Once again, these extremely readable, highly regarded, and widely adopted texts present "tried and true" formula pairing substantial ...show moreamounts of graphical analysis and informal geometric proofs with an abundance of hands-on exercises has proven to be tremendously successful with both students and instructors. What would the benefit to your students be of using a text which blends practical applications with mathematical concepts? Features
NEW - Details the ways in which technology can be used to foster understanding of several topics while it facilitates computation.
NEW - Ends each chapter with a Review of Fundamental Concepts, helping students focus on the chapter's key points.
NEW - Places greater emphasis on the significance of differential equations in applications involving exponential functions.
NEW - Customized calculus software is available through the study guide.
NEW - Companion website supports and extends the materials presented in the text.
NEW - All graphs of functions have been redrawn using Mathematicia.
Reinforces class lessons with carefully designed exercise sets, and challenges students to make their own connections.
Minimizes prerequisites, allowing those who have forgotten much of their high school mathematics to start anew with this self-contained material.
Functions and Their Graphs. Some Important Functions. The Algebra of Functions. Zeros of Functions. The Quadratic Formula and Factoring. Exponents and Power Functions. Functions and Graphs in Applications.
1. The Derivative
The Slope of a Straight Line. The Slope of a Curve at a Point. The Derivative. Limits and the Derivative. Differentiability and Continuity. Some Rules for Differentiation. More About Derivatives. The Derivative as a Rate of Change.
2. Applications of the Derivative
Describing Graphs of Functions. The First and Second Derivative Rules. Curve Sketching (Introduction.) Curve Sketching (Conclusion.) Optimization Problems. Further Optimization Problems. Applications of Calculus to Business and Economics.
3. Techniques of Differentiation
The Product and Quotient Rules. The Chain Rule and the General Power Rule. Implicit Differentiation and Related Rates.
Antidifferentiation. Areas and Reimann Sums. Definite Integrals and the Fundamental Theorem. Areas in the xy-Plane. Applications of the Definite Integral.
7. Functions of Several Variables
Examples of Functions of Several Variables. Partial Derivatives. Maxima and Minima of Functions of Several Variables. Lagrange Multipliers and Constrained Optimization. The Method of Least Squares. Double Integrals.
8. The Trigonometric Functions
Radian Measure of Angles. The Sine and the Cosine. Differentiation of sin t and cos t. The Tangent and Other Trigonometric Functions.
9. Techniques of Integration
Integration by Substitution. Integration by Parts. Evaluation of Definite Integrals. Approximation of Definite Integrals. Some Applications of the Integral. Improper Integrals
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3DSurface Viewer is a small Web application that creates high quality images of 3D surfaces defined by mathematical expressions. The quality of the images and the speed with which they are created ... More: lessons, discussions, ratings, reviews,...
The program uses the HTML5 canvas (HTML5 canvas javascript API) and web workers. Mathematical calculations are performed using the web workers, and the results are drawn on the canvas surface. Applica... More: lessons, discussions, ratings, reviews,...
This applet demonstrates an exponential growth model which plots population P_i for i=1 to i=600 given user input for the initial population P_0 and growth rate G. The difference equation used is P_(i... More: lessons, discussions, ratings, reviews,...
Using this virtual manipulative you may: graph a function; trace a point along the graph; dynamically vary function parameters; change the range of values displayed in the graph; graph multiple functiThis applet demonstrates a logistic growth model which plots population P_i for i = 1 to i = 600 given user input for the initial population P_0, growth rate G and carrying capacity CC. The difference... More: lessons, discussions, ratings, reviews,...
Enter a set of data points and a function or multiple functions, then manipulate those functions to fit those points. Manipulate the function on a coordinate plane using slider bars. Learn how each coPlomplex is a complex function plotter using domain coloring. You can compose a function with a complex variable z, and generate a domain coloring plot of it. You can choose the plot range as well as ... More: lessons, discussions, ratings, reviews,...
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Welcome to Algebra 2! There are no secrets to success. It is the result of preparation, hard work and learning from failure -Colin Powell- Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures So enter the doors of --- and lets explore the Wide World of Algebra 2. No need to guess; know your current grade or what you have missed by logging on to PINNACLE. Need your graduation status then VIRTUAL COUNSELOR is your information gateway. Need the latest news from your school? Why not log on to Miramar High and be informed.
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MAS 099. Presentation Attendance. The aim of this course is exposure to mathematics beyond the classroom curriculum. The course requirement is attendance at a minimum of six formal presentations on mathematical topics given at conferences, colloquia or symposia at a minimum of two separate events (that is, a conference or event). Presentations should have a title and abstract and may be given by faculty or students; poster sessions do not count.
0 credits.
MAS 100. Concepts of Mathematics. A study of a variety of topics in mathematics. Many introduce modern mathematics and most do not appear in the secondary school curriculum.
Fulfills general education requirement: Liberal Studies Area 4 (Mathematics).
3 credits.
MAS 102. Pre-Calculus. A review of precalculus mathematics including algebra and trigonometry.
A student may not receive credit for this course after completing MAS 111, 161, or the equivalent.
3 credits.
MAS 170. Elementary Statistics. An introduction to elementary descriptive and inferential statistics with emphasis on conceptual understanding.
Fulfills general education requirement: Liberal Studies Area 4 (Mathematics).
A student may not receive credit for MAS 170 after completing MAS 372. A student may not receive credit for both MAS 170 and MAS 270.
3 credits.
MAS 270. Intermediate Statistics. A more advanced version of MAS 170 intended for students with some calculus background.
Fulfills general education requirement: Liberal Studies Area 4 (Mathematics).
A student may not receive credit for both MAS 170 and MAS 270
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This is the revised and expanded second edition of the hugely popular Numerical Recipes: the Art of Scientific Computing. The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner, it proceeds from mathematical and theoretical considerations to actual, practical computer routines. With over 100 new routines, bringing the total to well over 300, plus upgraded versions of many of the original routines, this new edition is the most practical, comprehensive handbook of scientific computing available today. The book retains the informal, easy-to-read style that made the first edition so popular, even while introducing some more advanced topics. It is an ideal textbook for scientists and engineers, and an indispensable reference for anyone who works in scientific computing. The second edition is availabLe in FORTRAN, the quintessential language for numerical calculations, and in the increasingly popular C language.
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Algebra The Easy Way - 4th edition
Summary: The author describes and explains uses of equations, polynomials, the binomial formula, exponential functions, logarithms, and much more, and provides skill-building exercises with answers. Over the years, Barron's popular and widely used Easy Way books have proven themselves to be accessible self-teaching manuals. They have also found their way into many classrooms as valuable and easy-to-use textbook supplements. The titles cover a wide variety of practical and aca...show moredemic topics, presenting fundamental subject matter so that it can be clearly understood and providing a foundation for more advanced study. Easy Way books fulfill many purposes: they help students improve their grades, serve as good test preparation review books, and provide readers working outside classroom settings with practical information on subjects that relate to their occupations and careers. All new Easy Way editions feature type in two colors, the second color used to highlight important study points and topic headsBookBuyers Online1 CA San Jose, CA
2003
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Using History to Teach Mathematics An International Perspective
9780883851630
ISBN:
0883851636
Pub Date: 2000 Publisher: Mathematical Association of America
Summary: This is a volume of articles that provides insight, both in particular cases and in generality, into how the history of mathematics can find application in the teaching of mathematics itself. Educators at all levels, and mathematicians interested in the history of their subject, will find much of interest here.
Ships From:Boonsboro, MDShipping:Standard, ExpeditedComments:Brand new. We distribute directly for the publisher. This book is a collection of articles by in... [more] [[ and at university levels. Many of the articles can serve teachers directly as the basis of classroom lessons, while others will give teachers plenty to think about in designing courses or entire curricula. For example, there are articles dealing with the teaching of geometry and quadratic equations to high school students, of the notion of pi at various levels, and of linear algebra, combinatorics, and geometry to university students. But there is also an article showing how to use historical problems in various courses and one dealing with mathematical anomalies and their classroom use. Although the primary aim of the book is the teaching of mathematics through its history, some of the articles deal more directly with topics in the history of mathematics not usually found in textbooks. These articles will give teachers valuable background. They include one on the background of Mesopotamian mathematics by one of the world's experts in this field, one on the development of mathematics in Latin America by a mathematician who has done much primary research in this little known field, and another on the development of mathematics in Portugal, a country whose mathematical accomplishments are little known. Finally, an article on the reasons for studying mathematics in Medieval Islam will give all teachers food for thought when they discuss similar questions, while a short play dealing with the work of Augustus DeMorgan will help teachers provide an outlet for their class thespians.[less]
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SPI 3108.3.3 Describe algebraically the effect of a single transformation (reflections in the x- or y-axis, rotations, translations, and dilations) on two-dimensional geometric shapes in the coordinate plane.
CLE 3103.3.3 Analyze and apply various methods to solve equations, absolute values, inequalities, and systems of equations over complex numbers.
CLE 3103.3.4 Graph and compare equations and inequalities in two variables. Identify and understand the relationships between the algebraic and geometric properties of the graph.
CLE 3103.3.5 Use mathematical models involving equations and systems of equations to represent, interpret and analyze quantitative relationships, change in various contexts, and other real-world phenomena.
Checks for Understanding
3103.3.1 Perform operations on algebraic expressions and justify the procedures.
3103.3.2 Determine the domain of a function represented in either symbolic or graphical form.
3103.3.3 Determine and graph the inverse of a function with and without technology.
3103.3.4 Analyze the effect of changing various parameters on functions and their graphs.
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Another,perhaps somewhat odd andunconventionalProf.Björk being one of the best teachers I've had in terms of providing you with an intuitive feeling for the subject at hand, I'm sure that such sections would work very well for someone who is just trying to get anabasic understanding of the subject.Iamnotintimatelyfamiliarwiththebook(Iamnotdoingresearchinthatfield)butasIunderstandit,itisalmostrequiredreadingforanyonewhoisseriousaboutmathematicalfinance somewhat odd being one of the best teachers I've had in terms of providing you with an intuitive feeling for the subject at hand, I'm sure that such sections would work very well for someone who is just trying to get an understanding of the subject. I am not intimately familiar with the book (I am not doing research in that field) but as I understand it, it is almost required reading for anyone who is serious about mathematical finance.
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Freshman College Math
The mission of the Freshman College Math Department is to provide students with fundamental skills of mathematics and critical thinking so that they can make a successful transition to in higher-level classes.
The following information is offered to help you make decisions about our courses that will meet your mathematical needs as you prepare your academic plan for the future. Studies have shown that a diverse and broad background in mathematics expands opportunities in most career fields.
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Student Resources
Services
The courses offered by the Developmental Mathematics Department are designed to provide a foundation in preparatory mathematics necessary for success in future college courses throughout a variety of disciplines as well as mathematics. The courses also aim for the development of critical thinking skills applicable to all aspects of academic life.
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would review with the student the basic concepts in algebra I such as equations and inequalities, the coordinate system and functions of multiple variables; then examine in greater depth. Other topics will then include quadratic functions and factoring, polynomials; including complex numbers a
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Definition:
Algebra is the branch of math the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Commonly with geometry, analysis, topology, combinatory, and number theory, algebra is one of the main branches of pure mathematics.
Reference: Wikipedia, the free encyclopedia.
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Differential Equations For Dummies
The fun and easy way to understand and solve complex equations. Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores. Steven Holzner, PhD (Ithaca, NY), is the author of Physics For Dummies (978-0-7645-5433-9) and Physics Workbook For Dummies (978-0-470-16909-4). He taught physics at Cornell for over ten years and has written more than 95 books about programming.
Customer Reviews:
An Author that Truly Understands Keeping It Simple
By Terrance R. Banach "Terry the book addict" - July 14, 2008
Steven Holzner can truly make differential equations understood on a basic level. The author has an ability to keep you interested by keeeping the text light and easy to comprehend. He seems to understand what gives the typical student of diff-e-q difficulties and show a clean path right through these areas. Great book with simple answers to some of my past misunderstood solutions in differential equations.
Great value
By Dimitri Shvorob - June 30, 2009
Hopefully, nobody expects this Dummies book to match a $40+ college ODE textbook, not to mention a more advanced tome; my own expectations were modest, and the book was a pleasant surprise. I think it's not a bad idea to start one's study of ODE here; discussion of numerical methods in particular is a plus. (The small minuses are the sketchy discussion of the integrating- factor trick, and the derailed-by-typo solution on p. 111).
I'm not sure why you would want this
By Michael Dunipace - June 30, 2009
After having read the whole book, I must admit that it would be difficult for me to explain who would really want to read it. Not that it is a bad book. It covers some of the subjects you would get in a standard beginning Diff Eq class in college, but (as the title implies) just cannot do an adequate job given the size that it is. There is a reason that standard texts are as large as they are. And it has more typos than I would like, but again not so many as to be enough to stop reading it.
The real problem is with the subject matter itself. Writing a book for Diff Eq is not like writing "Golf for Dummies" or "Ebay for Dummies". Many topics that you don't know much about can be introduced in a light-hearted way with clear explanations and after a couple hundred pages and a few hours you can put the info to good use. This topic just isn't like that, in my opinion. You just can't get a lot out of a Diff Eq course without a fairly solid understanding of algebra and... read more
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Solutions and Dosage
ABSTRACT
An understanding of simple arithmetic, through "percentage" and "ratio and proportion," is essential to the accurate preparation and administration of drugs. This textbook provides a clear presentation of simple mathematical relations, explained in terms of both the apothecary and metric systems of units. It contains chapters on arithmetic review, on the preparation of solutions and on dosage, with many exercises and experiments designed to develop facility in accurate calculation. The problems deal with drugs used in modern therapy. This book should prove useful to student nurses, pharmacy students and physicians who wish to review simple arithmetic
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Schaum's has Satisfied Students for 50 Years.
Now Schaum's Biggest Sellers are in New Editions!,... more...
Trigonometry has always been the black sheep... more...
CliffsQuickReview course guides cover the essentials of your toughest classes. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions. CliffsQuickReview Trigonometry provides you with all you need to know to understand the basic concepts of trigonometry — whether you need a supplement to your... more...
Part of the ''Demystified'' series, this book covers various key aspects of trigonometry: how angles are measured; the relationship between angles and distances; coordinate systems; calculating distance based on parallax; reading maps and charts; latitude and longitude; and more. more...
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MAT
106
- Math for Elementary Education I
This is the first course of a two-semester sequence which explores the mathematics content in grades K-6 from an advanced standpoint. Topics include: problem solving; functions and graphs; and numbers and operations. This course is open to elementary education and early childhood students only.
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It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The importance of algebra, in particular, cannot be overstated, as it is the basis of all mathematical modeling used in applications found in all disciplines. Traditionally, the study of algebra is separated into a two parts, elementary algebra and intermediate algebra. This textbook, Elementary Algebra, is the first part, written in a clear and concise manner, making no assumption of prior algebra experience. It carefully guides students from the basics to the more advanced techniques required to be successful in the next course.
This text is, by far, the best elementary algebra textbook offered under a Creative Commons license. It is written in such a way as to maintain maximum flexibility and usability. A modular format was carefully integrated into the design. For example, certain topics, like functions, can be covered or omitted without compromising the overall flow of the text. An introduction of square roots in Chapter 1 is another example that allows for instructors wishing to include the quadratic formula early to do so. Topics such as these are carefully included to enhance the flexibility throughout. This textbook will effectively enable traditional or nontraditional approaches to elementary algebra. This, in addition to robust and diverse exercise sets, provides the base for an excellent individualized textbook instructors can use free of needless edition changes and excessive costs! A few other differences are highlighted below:
Equivalent mathematical notation using standard text found on a keyboard
A variety of applications and word problems included in most exercise sets
Clearly enumerated steps found in context within carefully chosen examples
Video examples available, in context, within the online version of the textbook
Robust and diverse exercise sets with discussion board questions
Key words and key takeaways summarizing each section
This text employs an early-and-often approach to real-world applications, laying the foundation for students to translate problems described in words into mathematical equations. It also clearly lays out the steps required to build the skills needed to solve these equations and interpret the results. With robust and diverse exercise sets, students have the opportunity to solve plenty of practice problems. In addition to embedded video examples and other online learning resources, the importance of practice with pencil and paper is stressed. This text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today. In addition, textual notation is introduced as a means to communicate solutions electronically throughout the text. While it is important to obtain the skills to solve problems correctly, it is just as important to communicate those solutions with others effectively in the modern era of instant communications.
Flat World Knowledge is the only publisher today willing to put in the resources that it takes to produce a quality, peer-reviewed textbook and allow it to be published under a Creative Commons license. They have the system that implements the customizable, affordable, and open textbook of the twenty-first century. In fact, this textbook was specifically designed and written to fully maximize the potential of the Flat World Knowledge system. I feel that my partnership with Flat World Knowledge has produced a truly fine example in Elementary Algebra, which demonstrates what is possible in the future of publishing.
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computer files ready to use
Click on any of the following links to download the file you are interested in using.
Choose "Save file to disk" if you want to put it on your desktop for temporary
use or keep it on your hard drive for future reference. You can also right-click
and choose "Save Target As..."
This Mathematica notebook is a palette you can
use to quickly graph a function. You can choose which options you want to include in your
code. It also has options for graphing multiple functions. Very easy to use!
The first is a short, simple program for creating
graphs with instructions
included. The second has a few programs for creating graphs of multiple functions and
graphs with legends and/or grid lines. I recommend playing with Graph Template 1
first. The third file allows you to adjust the
fonts of the labels and the colors of the grid lines too (including nice
values for trigonometry).
Graph one or two inequalities with shading. This Mathematica notebook
give you control over shading colors, solid and dashed lines, grid lines,
tick marks, axis labels, and graph title including fonts and colors. You can
also copy-and-paste graphs into other computer programs to make tests,
worksheets, or notes.
This Mathematica file lets you easily generate polar graphs
on a polar grid. Directions for inserting graphs into Word files are
also included. Try Tracing
a Polar Graph to create animations that draw the graphs.
Use this Mathematica notebook to
illustrate converting a point in rectangular coordinates to polar
coordinates. Copy-and-paste your diagrams into worksheets or tests.
(Note: There are some extra frills
with this one since I was using it to learn some new things.)
Ready to use for classroom demonstration. Two programs
for graphing "y = a cos[b(x - c)] + d." All
you need to do is decide which a, b, c, and d values
you want to see. It is easy to switch "cos" to "sin" or any other trig
function, if desired. The second program lets you animate the graph as any of the
parameters change between the values you set. Includes x-axis labels with p !
This Mathematica notebook is designed more for teachers than for
students. Generate problems quickly for students to solve systems of
equations problems by graphing, substitution, or linear combination. You
control the solution and the y-intercepts.
This lesson demonstrates the difference between
exponential and factorial growth. The lesson uses tables, graphs, and scientific notation.
It also includes directions for using Microsoft Word to answer the guide
questions throughout the lesson. This works for a classroom demonstration, computer lab
activity, or independent study. (Mathematica notebook file)
How do a cone and a plane really make an ellipse? What about
hyperbolas and parabolas? This Mathematica notebook allows you
to adjust the angle of a plane as it intersects a cone to see the curves
that can be generated. Also, see the Conic Sections Gallery below and the Blank
Conic Sections Diagrams.
This Mathematica notebook is a list
of 3D graphics that you can rotate and zoom to see these surface
intersections more clearly. Great for classroom demonstrations or for
students to explore. (Or check out the Interactive
HTML Version or the Still
HTML Version.)
Create figures generated by rotating a region about a
horizontal line. You can specify the region using two functions and you can also pick the
axis of rotation. This is great for illustrating disk- and washer-method problems. It's
very easy to use.
Choose your own region to rotate about the y-axis
and see it formed by a series of cylinders. This program nicely illustrates the shell
method for finding volumes. Instructions are also included for a simple animation.
This file is a catalog of several different
three-dimensional figures created with a given base and a given cross sectional shape.
Also, the four programs I wrote to generate these figures are available to download.
You
can create your own!Create solids with square,
equilateral triangle, semicircle, or rectangle cross sections. 3D Web Graphics!
Directions and links on Interactive
Graphics page.
This Mathematica notebook is a palette to help
you graph curves in three dimensions using parametric equations. It also includes several
options for working with 3D graphs. (Designed for a lesson with Calculus D.)
This program is set up for graphing quadric
surfaces such as the example:
+ = 1. Instructions are included. Be sure to read the hints at the bottom in
case your results don't come out as you expected. Or check out this simple Mathematica
notebook with interactive
examples of each quadric surface.
This PDF file is a lesson designed to show you how to work with series in
Mathematica and to improve your understanding of infinite series. To
download the example code for the third part, click here.
To easily compare the graph of a sequence and the corresponding series, use SequenceSeriesPlot.
Create slope fields
with this Mathematica notebook. Draw curves passing through initial
points and curves corresponding to fixed values of C. Other options
are also included. It explicitly solves the differential equation. For some
functions, there will be errors.
Notebook with Only Slope Fields.
Limits - Illustrations with Geometer's Sketchpad
For each of the four cases below, you can
interactively illustrate the formal definitions of limits. (Also, check
out the Worksheet
and the single file containing
all four sketches.)
This PDF file contains Mathematica code for drawing vector fields
in two and three dimensions. It also guides the user through graphing a
surface and its level curves with its gradient field. The four-dimensional
equivalent is also included.
Curvature is a measure of how tightly a curve is turning. This Mathematica
notebook allows you to type your own 2-dimensional function and select
the point where you want to see the circle whose curvature matches the
curve at that point. Thanks to Whitney Buchanan (TP '03) for
inspiring and helping with this one!
Here is a
calendar just for TPHS and LCCHS
teachers and students. You can
see the entire semester for a class on one page. White means you meet that day, gray means
you don't. Great for planning, notes, and reference.(Updated:
August 11, 2012)
Assignment Sheets
Here are some blank assignment sheets students can use
to keep track of their work for a single class or for a week. Print
one page for each class or each week. Find the one that works best for
you.
It's not a "To Do List," it's
a "To Do Matrix"! This one page has helped keep me organized
for years. Slightly altered versions also work well for organizing
activities in class and appointments with students. To download the PDF version, click
here.
Blank Graph Paper
Six blank graphs per page ready to use for class activities and
homework. Polar
Graphs(Now with 4 different styles!) or Rectangular
Graphs (PDF files)
3D Graph Paper
Blank axes for 3D
graphing created by a student. (Thanks, Austin Landow!)(If the Word
version doesn't open, try right-click and "Save Target
As...")
Are you frustrated with scissors and tape to
cut-and-paste graphs into your tests, quizzes, and worksheets? Download this file and you
can electronically resize, cut, copy, and paste blank and numbered graphs into any
document you're working on. Very handy!
One page with a blank parabola, ellipse, and
hyperbola. Each one shows the directrix, center, foci, and vertices
without any numerical values or labels. Have students fill it in a study
guide, insert the diagrams into
tests and worksheets, or print
a copy on a transparency to use during classroom demonstrations.
(Note: It works well for labels, but the scale is slightly off for
comparing focus, curve, and directrix measurements. Look for updates
soon.)
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Merchandising/Retail Buying
Featured Products In Merchandising/Retail BuyingMerchandising math is a multifaceted topic that involves many levels of the retail process,
including assortment planning; vendor analysis; markup and pricing; and terms of sale.
A Practical Approach to Merchandising Mathematics, Revised 1st Edition, brings each of
these areas together into one comprehensive text to meet the needs of students who will be
involved with the activities of merchandising and buying at the retail level. Students will learn
how to use typical merchandising forms; become familiar with the application of computers
and computerized forms in retailing; and recognize the basic factors of buying and selling
that affect prof it. View
The recipe for profitability is presented in Merchandise Planning Workbook. Focusing on the development of a six-month merchandise plan, the text explains how to use Excel 2007 as a tool to project sales, manage inventory, calculate the amount of merchandise to purchase, and adjust the price throughout the selling season. Application Exercises throughout the chapters familiarize students with each aspect of the plan, provide practice in inputting formulas and data, and demonstrate the impact of changing variables. Seven end-of-chapter assignments, when completed in sequence, produce a merchandise plan for a selling season. By mastering this important aspect of merchandising math, students can develop a marketable competency to help launch their careers in retailing. View
The 3rd edition of this classic text continues to use the fictitious Perry's Department Store to bridge the gap between the principles of retail buying and mathematical formulas and concepts. The authors use their experience to provide students with the tools to understand a buyer's responsibilities by walking them through the various steps a new buyer would take to complete a six-month dollar plan and a merchandise assortment plan. This new edition emphasizes the professional perspective with the inclusion of two new chapters that go beyond theory to explain the importance of preparation before the buyer enters the market. View
Merchandising: Theory, Principles, and Practice, 3rd Edition, focuses on the process of merchandising and the principles applied to the planning, development, and presentation of product lines in both the manufacturing and retailing sectors. Each chapter includes case studies that illustrate how merchandising principles and theories are applied by actual businesses, and the chapter learning activities promote an interactive learning environment with multiple course objectives. Students will learn how to make sequential and integrated decisions to develop a complete merchandise plan and analyze the effectiveness of that plan. View
This textbook prepares future retail executives for the challenges they will face in contemporary retailing and manufacturing. Concepts and Cases in Retail and Merchandise Management, 2nd Edition, includes more than 70 cases that are contextualized by clear introductions and give students a grounding in a wide variety of contemporary retail management challenges. Case studies explore topics ranging from how to position a store and its merchandise to how to safeguard against cheap imports that threaten domestic manufacturers. View
The area of retail buying relies heavily on mathematical formulas and forecasting. The formulas themselves remain unchanged. However, the context in which they are analyzed is constantly evolving. The most successful retail buyers are able to withstand the highs and lows of business trends by utilizing analytical skills, trend forecasting, and customer knowledge. As a retail buyer for almost 20 years with various retailers, Connell draws on her experience providing practical fundamental mathematical formulas while also giving context in the current retail environment. This text gives students a step-by-step approach to understanding the mechanics of a six-month merchandising plan. Upon completion of the plan students will gain insight into how the plan is forecast into the future as well as how it is implemented at the actual purchasing level. View
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Mathematical Literacy
Mathematical Literacy is offered as a Matriculation subject from Form III at St Mary's. The subject focuses on real-life situations and is perfect for the pupil who finds difficulty with the more abstract facets of core Mathematics. Since Mathematical Literacy deals with real-life situations, it enables pupils to approach problems with confidence and allows them to discover methods that work for them. It equips pupils with techniques, knowledge, skills, values and attitudes for self-fulfilment and growth so that they can be meaningful participants in society. Mathematical Literacy lends itself successfully to activities such as presentations, research, debates, interviews, questionnaires, project work, assignments and practical investigations. Mathematical Literacy is accepted at university faculties where core Mathematics is not a prerequisite.
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From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
Description
This text addresses the need for a new mathematics text for careers using digital technology. The material is brought to life through several applications including the mathematics of screen and printer displays. The course, which covers binary arithmetic to Boolean algebra, is emerging throughout the country and may fill a need at your school. This unique text teaches topics such as binary fractions, hexadecimal numbers, and Venn diagrams to students with only a beginning algebra background.
Table of contents
1. Computation.
An Introduction.
Exponents and Their Properties.
Calculator Functions.
Scientific Notation.
An Introduction to Statistics and Error Analysis.
Dimensional Analysis.
2. Binary Numbers.
The Binary System.
Base Two Arithmetic.
Two's Complement.
Binary Fractions.
Computer Memory and Quantitative Prefixes.
3. Octal and Hexadecimal Numbers.
The Octal System.
Hexadecimal Representation.
Base 16 Arithmetic.
Elements of Coding.
4. Sets and Algebra.
The Language of Sets.
Set Operators.
Venn Diagrams.
Propositions and Truth Tables.
Logical Operators and Internet Searches.
5. Boolean Circuits.
Equivalent Boolean Expressions.
Logic Circuits Part I - Switching Circuits.
Truth Tables and Disjunctive Normal Form.
Logic Circuits Part II - Gated Circuits.
Karnaugh Maps.
6. Graphs.
Color Sets.
Hexadecimal RGB Codes.
Cartesian and Monitor Coordinates.
Elements of Computer Animation.
Features & benefits
Practice Problems Computer math students require a large number of exercises in order to practice the material that they have just learned. These exercises actively engage students in the learning process and reinforce newly learned concepts and skills.
Bits of History Scattered throughout the text, Bits of History let students in on the background and the origins of the concepts they are currently learning. This feature helps students better understand the theories behind their actions.
Definition Boxes Important definitions are boxed and highlighted throughout each chapter to emphasize their importance to students and to make them easy to find for review purposes.
Elementary Logic More than any other text offered at this level, this text emphasizes the application of logic to programming and circuitry. Logic is part of the foundation of what the text refers to as its "Rosetta Stone." By the end of the course, students see that a logic circuit can be represented by a truth table, a Boolean expression, or a Venn diagram.
Ample opportunity for Review At the end of each chapter is a comprehensive review section. Starting with a chapter Summary, students are given the main concepts of the chapter. The Glossary is a comprehensive listing of key terms from throughout the chapter. Review Exercises require students to solve problems without the help of section references. Additionally challenging problems are highlighted by a triangle symbol around the exercise number. Cumulative Review Exercises gather various types of exercises from the preceding chapters to help students remember and retain what they are learning throughout the course.
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Courses
Course Details
MATH 038 Pre-Algebra and Study Skills
4
hours lecture,
4
units
Letter Grade or Pass/No Pass Option
Description: This course is a study of the fundamentals of arithmetic operations with signed numbers, including fractions and decimals as well as an introduction to some elementary topics in beginning algebra. Topics also include ratios and proportions, perfect squares and their square roots, elementary topics in geometry, systems of measurement, and monomial arithmetic.
Students learn basic study skills necessary for success in mathematics courses. This course is intended for students preparing for Beginning Algebra.
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GeoGebra is dynamic math software that can be used in the study of algebra, geometry, and calculus. It is free and can either be run inside a web browser, or downloaded and installed locally for offline use. I have been using GeoGebra with my algebra and geometry students for several years, but recently discovered its usefulness for my discrete math students as well. As a culminating project for our unit on graph theory, I ask students to create and name their own graph. We first look at the Gallery of Named Graphs to brainstorm ideas, and then I let them loose to create a unique graph of their own. They have to describe the characteristics of their graph, using the concepts we've studied throughout the unit such as:
Students then present their graphs to the class and ask the audience questions about its characteristics. Rather than simply drawing a picture of their graph on paper, students create their graphs in GeoGebra and export them as interactive web pages. This allows the presenter to manipulate the graph during the presentation. It also allows the presentation to be interactive. Volunteers can come forward and try to show that the graph is planar, for example.
2 thoughts on "Investigating Graph Theory with GeoGebra"
Hello Mrs. K,
I'm glad to see you have a WordPress site with GeoGebra apps embedded.
I have not been able to embed apps on my WordPress site.
Could you let me know how you have been able to do this please.
Thanks,
Bill Lombard
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Description: The main goal of this course is to discuss Galois theory, which is the study of relationships among roots of polynomials. For example, we will use Galois theory to prove that there is no formula analogous to the quadratic formula for the roots of xn - x - 1 when n is at least 5, or in fact for the roots of most polynomials of degree at least 5. More generally, Galois theory provides a correspondence between two different topics in algebra: fields and groups. Our study of fields will use linear algbera in interesting ways. For example, we will see how to show certain polynomials are irreducible using the concept of dimension. Only towards the end of the course will group theory be needed in a serious way, at which point what we need from group theory will be reviewed. Prerequisites: Math 3230(216). Offered: Spring (odd years) Credits: 3
These are the most recent data in the math department database for Math 3231 in Storrs Campus.
There could be more recent data on our class schedules page, where you can also check for sections at other campuses.
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Welcome to Math 251 - Online!
Below you will find answers to common questions about this course:
Learning in an Online Environment
The best part about online learning is it offers flexibility. You can choose when and where you want to learn and you don't have to fight traffic or find parking to get there. You just need your computer and a great attitude.
Online learning is not for everyone. Ask yourself if you are disciplined and motivated enough to take a few hours a day, 3 or 4 times a week, to succeed in learning algebra and to pass this class. There will be no class that you have to be in at a certain time and no professor right there in class to answer your question.
What you will have is MyMathLab. MyMathLab has taken your textbook and put it online. It has enhanced it with buttons you can click to get short audio and video clips, see animations, and try practice problems with guided solutions. Each section of the book begins with a video lesson explaining the material. While doing the homework if you get stuck, MyMathLab can help you. It can walk you through examples, give you solutions, generate new similar problems and even take you to the part of the video that explains the particular type of problem you are working on. MyMathLab also has an online tutoring center.
The only time you are required to come to campus is for the final exam. All other work is online.
What if I can't make the on-campus final exam?
If you are out of the Southern California area and coming to the final exam would create a hardship, you must arrange a proctor through another college, high school, library, or military educational officer. Proctors need to be arranged at least 2 weeks before the scheduled final exam date and your final exam must be completed by the scheduled final exam date. A list of colleges that proctor for a small fee is available at
Course Website/Online Textbook
For those of you who have taken online courses with the college before, this one is not via the Saddleback College Blackboard site. Instead, the course uses MyMathLab, a website maintained by the textbook's publisher. Intead of buying a hard copy of the textbook, I require each student to register for this site which includes the online textbook. The cost of registering is about 1/2 the price of the hard copy book. The orientation assignment will give you instructions on how to register.
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Search
Students will approximate the area under a curve using Riemann sums. This will be done by utilizing a program that computes the Riemann sum as well as drawing the graphical representation. The activity concludes with students discovering that if enough Riemann sums are used, then the area under a curve can be calculated with the required degree of precision.
From Sean Bird's website: "Put this [TI-Nspire] file in MyLib so that you can access the area approximation methods from any document."
The program will find the approximate areas for the left, right and midpt Reimann sums, as well as the trapezoid, Simpsons and numeric integral areas.
A lovely Geogebra java applet from Miguel Bayona, The Lawrenceville School, Lawrenceville, NJ. It demonstrates the lower sum, upper sum, left sum, right sum, midpoint sum and trapezoidal sum for a function of your choosing. Loading the applet can take a while, so be patient.
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Hi, can anyone please help me with my math homework? I am not quite good at math and would be grateful if you could explain how to solve mathematical mcqs problems. I also would like to find out if there is a good website which can help me prepare well for my upcoming math exam. Thank you!
Can you please be more descriptive as to what sort of guidance you are expecting to get. Do you want to learn the fundamentals and solve your math questions by yourself or do you need a utility that would provide you a step-by-step solution for your math problems?
I must agree that Algebrator is a great thing and the best program of this kind you can get. I was amazed when after weeks of frustration I simply typed in binomial formula and that was the end of my problems with math. It's also so good that you can use the program for any level: I have been using it for several years now, I used it in Basic Math and in Pre Algebra also! Just try it and see it for yourself!
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TEXTBOOK*
Graph Theory: A Problem Oriented Approach
Daniel A. Marcus
Can be used as a college-level text for mathematics, computer science or engineering students. Also suitable for a general education course at a liberal arts college, or for self-study problem-oriented format is intended to promote active involvement by the reader while always providing clear direction. This approach figures prominently on the presentation of proofs, which become more frequent and elaborate as the book progresses. Arguments are arranged in digestible chunks and always appear along with concrete examples to keep the readers firmly grounded in their motivation.
Spanning tree algorithms, Euler paths, Hamilton paths and cycles, planar graphs, independence and covering, connections and obstructions, and vertex and edge colorings make up the core of the book. Hall's Theorem, the König-Egervary Theorem, Dilworth's Theorem and the Hungarian algorithm to the optimal assignment problem, matrices and latin squares are also explored.
* As a textbook, Graph Theory does have DRM. Our DRM protected PDFs can be downloaded to three computers.
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The Chemistry Maths Book provides a complete course companion suitable for students at all levels. All the most useful and important topics are covered with numerous examples of applications in chemistry and the physical sciences.
Presents a set of answers to the exercises in Junior Maths 3. This text is a complete set of answers to the exercises in Junior Maths 3. Junior Maths 3 is ideal for Key Stage 2 maths pupils and is suitable for use from Year 5.
Presents an introduction to maths for Key Stage 2 pupils. Suitable for use from Year 3, this book provides a foundation for mathematical study, covering the basics of addition, subtraction, multiplication and division. Junior Maths 1 is the...
Providing numeracy practice for Year 5 of Primary School this Maths Workout book contains several quick-fire questions to test children's knowledge. It is suitable for working through at home to supplement school numeracy lessons and reinforce children's understanding of topics.
Produced in partnership with OCR, University of York Science Education Group and Nuffield Foundation, these second editions of the Twenty First Century Science resources provide the best support for the new specifications and make the transition as smooth as possible. This pack provides the support needed to teach the new 2011 specifications.
This book is a collected series of the popular teaching articles from the open-access, online journal CBE--Life Sciences Education. The Allen-Tanner essays are practical guides that share insights and strategies for teaching science...
Suitable as a resource for teachers to use alongside So You Really Want to Learn Maths Book 1. This work features photocopiable worksheets that can be used to complete selected exercises (indicated by the worksheet symbol in the textbook) featured in So You Really Want to Learn Maths Book 1.
This text is a comprehensive introduction to the use of models and modeling in science education. It identifies and describes many different modeling tools and presents recent applications of modeling as a cognitive tool for scientific enquiry.
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Combinatorial Introduction To Topology (94 Edition)
by Michael Henle Publisher Comments
Excellent text for upper-level undergraduate and graduate students shows how geometric and algebraic ideas met and grew together into an important branch of mathematics. Lucid coverage of vector fields, surfaces, homology of complexes, much more. Some... (read more)
College Algebra (5TH 11 Edition)
by Mark Dugopolski Publisher Comments
This package consists of the textbook plus an access kit for MyMathLab/MyStatLab. Dugopolski's College Algebra, Fifth Edition gives readers the essential strategies to help them develop the comprehension and confidence they need to be... (read more)
Linear Algebra With Applications (6TH 02 - Old Edition)
by Steven J. Leon Publisher Comments
This thorough and accessible book from one of the leading figures in the field of linear algebra provides readers with both a challenging and broad understanding of linear algebra. The author infuses key concepts with their modern practical... (read more)
Testing Structural Equation Models (93 Edition)
by Kenneth A. Bollen Synopsis
What is the role of fit measures when respecifying a model? Should the means of the sampling distributions of a fit index be unrelated to the size of the sample? Is it better to estimate the statistical power of the chi-square test than to turn to fit... (read more)
Statistical Inference (2ND 02 Edition)
by George Casella Publisher Comments
This book builds theoretical statistics from the first principles of probability theory. Starting from the basics of probability, the authors develop the theory of statistical inference using techniques, definitions, and concepts that are statistical and... (read more)
Glencoe Mathematics Geometry (04 Edition)
by Cindy J. (ed.) Boyd Publisher Comments
A flexible program with the solid content students need Glencoe Geometry is the leading geometry program on the market. Algebra and applications are embedded throughout the program and an introduction to geometry proofs begins in Chapter 2.... (read more)
Numerical Analysis -study Guide (9TH 11 Edition)
by Richard L. Burden Publisher Comments
The Student Solutions Manual and Study Guide contains worked-out solutions to selected exercises from the text. The solved exercises cover all of the techniques discussed in the text, and include step-by-step instruction on working through the algorithms.... (read more)
Mathematical View of Our World - With CD (07 Edition)
by Harold Parks Publisher Comments
Harness the power of mathematics in school and your future career with A MATHEMATICAL VIEW OF OUR WORLD. This liberal arts textbook helps you see the beauty and power of mathematics as it is applied to the world around you. You will recognize the... (read more)
Numerical Methods With Matlab (00 Edition)
by Gerald Recktenwald Publisher Comments
This book is an introduction to MATLAB and an introduction to numerical methods. It is written for students of engineering, applied mathematics, and science. The primary objective of numerical methods is to obtain approximate solutions to problems that
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Mathematical Induction
In this lesson our instructor talks about mathematical induction. He answers the question what is induction and does examples of mathematical induction. He talks about the historical background of induction where he discusses the French mathematician Pierre de Fermat. He also talks about Leonhard Euler. Lastly, he talks about the principle of mathematical induction. Four extra example videos round up this lesson.
This content requires Javascript to be available and enabled in your browser.
Mathematical Induction
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
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Elementary Geometry - 3rd edition
ISBN13:978-0471510024 ISBN10: 0471510025 This edition has also been released as: ISBN13: 978-0471537465 ISBN10: 0471537462
Summary: Although extensively revised, this new edition continues in the fine tradition of its predecessor. Major changes include : A notation that formalizes the distinction between equality and congruence and between line, ray and line segment; a completely rewritten chapter on mathematical logic with inclusion of truth tables and the logical basis for the discovery of non-Euclidean geometries ; expanded coverage of analytic geometry with more theorems discussed an...show mored proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms ; a condensed chapter on numerical trigonometry; more problems; expansion of the section on surface areas and volume; and additional review exercises at the end of each chapter. Concise and logical, it will serve as an excellent review of high school geometry18.00 +$3.99 s/h
Acceptable
Nettextstore Lincoln, NE
1991 Hardcover Fair CONTAINS SLIGHT WATER DAMAGE / STAIN, STILL VERY READABLE, SAVE$$! !
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Search Course Communities:
Course Communities
Demos for Max-Min Problems
Instructors' notes break down steps for illustrating fundamental concepts for understanding and developing equations that model optimization problems, commonly referred to as max-min problems. The focus is on geometrically based problems so that animations can provide a foundation for developing insight and equations to model the problem. The common max-min problems illustrated include the following: "Maximize the area of a pen," "Minimize the time for rowing and walking," "Maximize the volume of an inscribed cylinder," "Maximize the area of an inscribed rectangle," "Determine the point on a curve closest to a fixed point," "Maximize the area for two pens," "Maximize the area of a rectangle inscribed in an isosceles triangle," "Maximize the printable region of a poster," "Construct a box of maximum volume," "Construct a cone of maximum volume," "Maximize the viewing angle of the Statue of Liberty," and "Minimize the travel time for light from one point to another."
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Algebra today, in the author's words \lq\lq plays a critical role in the development of the computer and communication technology that surround us in our daily lives." The goal of this undergraduate book is \lq\lq to show that the subject is alive, vibrant, exciting, and more relevant to modern technology than it has ever been." Accordingly the book presents the classical algebraic topics (modular arithmetic, groups, rings, fields and algebraic geometry) with a view on their applications to Computer Science, Error Correcting Codes and Cryptography and sections devoted to those applications are included through the book. Chapter 1 studies modular arithmetic with Section 1.8 devoted to bar codes and ISBN and Section 1.12 introducing the idea of public key cryptography. Chapter 2 provides the basic ideas about rings and fields, Section 2.14 studying error correcting codes (including BCH and Reed-Solomon codes). Chapters 3 and 4 deal with group theory and Section 4.10 illustrates the topic of substitution cipher with the description of the Enigma machine. Chapter 5 studies rings of polynomials, Gröbner bases and affine varieties. The last two chapters are devoted to elliptic curves. Chapter 6 introduces the concept of elliptic curve and its group law and states some classical results concerning elliptic curves over $\mathbb{B}$\, (theorems of Mordell-Weil, Mazur and Lutz-Nagell) and over a finite field (Hasse's theorem). Chapter 7 gathers some further topics related to elliptic curves as elliptic curve cryptosystems, the role of elliptic curves in the Wiles' proof of the Fermat last theorem or the Lenstra's factoring algorithm (the author also points out the existence of elliptic primality tests but he do not detail them). As conclusion, this book combines an introduction to abstract algebra with the presentation of some of its modern technological applications, which can contribute to awake up the interest of the students for this branch of the mathematics.
Reviewer:
Juan Tena Ayuso (Valladolid)
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Note to Readers: This is the original version of the Math Maturity page. On 10/6/09 it was renamed to Math Maturity v1 and a new Math Maturity page was created. While the two pages overlap considerably, the newer page has been heavily redesigned and edited. Most readers will likely decide the new version is superior to the older one.
"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, The Disciplined Mind: What All Students Should Understand, Simon & Schuster, 1999.)
"If we desire to form individuals capable of inventive thought and of helping the society of tomorrow to achieve progress, then it is clear that an education which is an active discovery of reality is superior to one that consists merely in providing the young with ready-made wills to will with and ready-made truths to know with…" (Jean Piaget; Swiss philosopher and natural scientist, well known for his work studying children, his theory of cognitive development; 1896–1980.)
Note to Readers
Math maturity is a complex topic. This article represents my explorations and understanding of the topic. It is a relatively long and convoluted article, drawing on a number of reference sources.
After reading the Introduction, you might want to jump to the Math Maturity section, thereby skipping much of the discussion of the research literature and more quickly arriving at some conclusions and recommendations.
Introduction
The three quotes given above help to give the flavor of this document. Math is a very important discipline both in its own right and because of its widespread applications in other disciplines.
Many people believe that the math education system in the United States is not nearly as effective as it should be. Over the years, there have been considerable efforts to improve the effectiveness of our math education system. Many of these efforts have focused on developing better curriculum and books, providing better preservice and inservice education for math teachers, requiring more years of math courses for precollege students, and setting more rigorous standards. There has also been a strong emphasis on encouraging women and minorities to take more math.
Current efforts to improve our math education system tend to be mostly focused on the same approaches. The general feeling seems to be that if we can just do more and better in these approaches, our math education system will improve.
Here are four important areas that have received much less attention.
There is substantial and mounting evidence that the math education curriculum in the United States is not designed and taught in a manner consistent with what is known about math cognitive development of students. Research in brain science is progressing more rapidly than our implementation of the results in our educational systems.
Our current math education system is not nearly as successful as we would like in helping students gain in their math creativity knowledge and skills, in their ability to attach and solve novel, complex, challenging problems, and in their ability to transfer their math knowledge and skills to problems outside the discipline of mathematics.
Our current math education system is still rather weak in teaching and learning in a manner that appropriately deals with forgetting. We know that students in math classes eventually (or, quite quickly) forget much of what they supposedly have learned. Although we spend quite a bit of time on review, we still face the constructivist problem that we are expecting students to build (construct) new knowledge on top of knowledge that they do not have.
In recent years, math educators have also had to deal with the steadily increasing capability and available of calculators and computers. In essence, we now need an education system that deals with both human brains and computer brains, and how to prepare humans to work effectively in environments where the computer capabilities increase significantly year by year.
There are many different approaches to the study of math education and in exploration of ways to improve our math eduction system. This document explores math intelligence, math cognitive development, and math maturity. It includes a focus on how to help students increase their level of math maturity.
In very brief summary, math maturity consists of an appropriate combination of math knowledge and skills, and the ability to think, understand, and solve problems using the math knowledge and skills. With disuse over time, one forgets much of their "learned" math knowledge and skills. However, one's level of math maturity—one's level of math-oriented thinking, understanding, and problem-solving—tends to have long term retention. Much more detail about math maturity is provided later in this document.
As with other documents in this IAE-pedia, the goal is to help improve education at all levels and throughout the world. Readers need to keep in mind that there are many different approaches to improving math education.
Marshmallows and Delayed Gratification
Quite a bit of formal education involves delayed gratification. This is certainly true in math education. When a student asks, "Why do I need to learn this?" a frequent response is, "You are going to need it next year." Of course, an common response nowadays is also, "It is going to be on the test." Personally, I find such a response rather unsatisfactory.
There has been some amusing and interesting research on a type of delayed gratification of young children. You can read a New Yorker magazine article on this, or view the short video on the test.
Youngsters are tested on whether they can delay eating a marshmallow (or some other "treat") in order to get two of the treats 15 minutes later. Only about 1/3 of the four-year old US children in the original research and 1/3 of the 4–6 year old Colombian children in research on children in that country were able to delay for 15 minutes.
Follow-up research on the US children 15 years later indicated that all who were able to delay their gratification for 15 minutes had been quite successful as students and in other parts of their lives.
Here is a math education quote from the New Yorker article:
Angela Lee Duckworth, an assistant professor of psychology at the University of Pennsylvania, is leading the program. She first grew interested in the subject after working as a high-school math teacher. "For the most part, it was an incredibly frustrating experience," she says. "I gradually became convinced that trying to teach a teen-ager algebra when they don't have self-control is a pretty futile exercise." And so, at the age of thirty-two, Duckworth decided to become a psychologist. One of her main research projects looked at the relationship between self-control and grade-point average. She found that the ability to delay gratification—eighth graders were given a choice between a dollar right away or two dollars the following week—was a far better predictor of academic performance than I.Q. She said that her study shows that "intelligence is really important, but it's still not as important as self-control." [Bold added for emphasis.]
Background on Innate Human Math Capabilities
There is research backing the idea that several month old human babies have innate ability to recognize small quantities, such as noticing that there is a difference between two of something and three of that thing. A variety of other animals have somewhat similar innate sense of quantity.
Although fractions are thought to be a difficult mathematical concept to learn, the adult brain encodes them automatically without conscious thought, according to new research in the April 8, 2009 issue of The Journal of Neuroscience. The study shows that cells in the intraparietal sulcus (IPS) and the prefrontal cortex - brain regions important for processing whole numbers - are tuned to respond to particular fractions. The findings suggest that adults have an intuitive understanding of fractions and may aid in the development of new teaching techniques.
"Fractions book The Math Gene (Devlin, 1999) presents an argument that the ability to learn to speak and understand a natural language such as English is a very strong indication that one can learn math.
In essence, Devin argues that a student's development of math knowledge and skills is mostly dependent on informal and formal education coming from parents, teachers, television, games, and so on. See his opening keynote presentation at the 2004 NCTM Annual Conference.
This insight helps us to understand one of the major challenges in our current math education system. A great many parents were not particularly successful in learning math, and typically they do not provide a "rich" math environment for their children. A great many elementary school teachers are not particularly strong in math. The math environments they provide in their classrooms tends to consist of "covering" the math book and its related curriculum. Their level of math maturity is modest, as is their interest in and enthusiasm for math.
AS a consequence of this, many young students do not gain nearly as high a level of math maturity as they might. This is not a consequence of their innate math abilities. Rather, it is a consequence of the informal and formal math education that they receive at home, in their community, and in their early years of schooling.
Intelligence and Intelligence Quotient (IQ)
"Did you mean to say that one man may acquire a thing easily, another with difficulty; a little learning will lead the one to discover a great deal; whereas the other, after much study and application, no sooner learns than he forgets?" (Plato, 4th century B.C.)
As the quote from Plato indicates, people have long been interested in intelligence. It has long been known thatpeople vary considerably in their rate and quality of their learning.
There is substantial research to support the contention that students of higher IQ learn faster and better than students of lower IQ. A teacher in a typical elementary school classroom may have one or two students who can learn twice as fast (and better) than the average students in the class, and one or two who learn half as fast (and not as well) as compared to the average students in the class.
Here is a little more recent history on measuring IQ. Quoting from the Wikipedia:
The Stanford-Binet test started with [the 1904 work of] the French psychologist Alfred Binet, whom the French government commissioned with developing a method of identifying intellectually deficient children for their placement in special education programs. As Binet indicated, case studies might be more detailed and helpful, but the time required to test many people would be excessive.
Later, Alfred Binet and physician Theodore Simon collaborated in studying mental retardation in French school children. Theodore Simon was a student of Binet's. Between 1905 and 1908, their research at a boys school, in Grange-aux-Belles, led to their developing the Binet-Simon tests; via increasingly difficult questions, the tests measured attention, memory, and verbal skill. Binet warned that such test scores should not be interpreted literally, because intelligence is plastic and that there was a margin of error inherent to the test (Fancher, 1985). The test consisted of 30 items ranging from the ability to touch one's nose or ear when asked to the ability to draw designs from memory and to define abstract concepts. Binet proposed that a child's intellectual ability increases with age. Therefore, he tested potential items and determined that age at which a typical child could answer them correctly. Thus, Binet developed the concept of mental age (MA), which is an individual's level of mental development relative to others.
A driving force in Binet's work and the work of others in the field of IQ is the goal of developing a measurement that is reasonably accurate in predicting future success in school, work, and other cognitive-related activities. For example, with accurate information one can better align formal schooling with the cognitive abilities of a student, and one can better advise a student about informal and formal academic and career choices.
Notice the initial measures of intelligence measured attention, memory, and verbal skill. An ADHD student might do poorly on such a test. A child growing up in a "rich" verbal environment will tend to score much better on such a test than a child growing up in a poorer oral communication environment.
Here are four important ideas that have been developed and/or more fully explored since the initial work of Binet:
A typical person's brain reaches full maturity at approximately 25 to 27 years of age. One's brain continues to change significantly over the years, showing a marked level of plasticity. Thus, continuing active use and education of one's brain can maintain and continue to improve its level of performance for a great many years after its full physical maturity is reached.
As Binet pointed out, a child's intellectual ability increases with age. This led researcher to "norm" the scores on intelligence tests, in an attempt to produce a number (called IQ) that remains relatively stable over time.
Many researchers have explored the idea of a single general intelligence factor called "g" versus multiple intelligences. There is a quite high level of correlation between "g" and the various multiple intelligences identified by Howard Gardner, Robert Sternberg, and other researchers who have focused on the general area of multiple intelligences.
In terms of the fourth point given above, logical/mathematical is one of the eight multiple intelligences identified by Gardner. Creativity is one of the three multiple intelligences identified by Sternberg. From the point of view of these two theories of multiple intelligences, learning to make effective use of one's innate math abilities and learning to make creative mathematical use of one's brain are ways to increase one's level of math maturity.
Measuring Intelligence
There are many different ways to attempt to measure intelligence. It turns out that this is a very challenging task. This is further complicated by the fact that intelligence is strongly influenced by both nature (one's genetic makeup) and nurture (informal and formal education and life experiences). The "nurture" component of intelligence is also affected by things like the quality of food one eats (starvation is bad for the brain), poisons (mercury and lead are bad for the brain), brain injuries (brain damage can severely disrupt a brain's capabilities), and so on.
Have you ever wondered why an average person has an IQ of about 100, and that this often does not change much over the years? Surely an average person develops quite a bit mentally as he or she grows from infancy to adulthood and learns a great deal during this time through informal and formal education and through life experiences.
The explanation to this situation lies in the way that intelligence is measured and reported. Measures of intelligence are usually normed in a manner that makes one's IQ a relatively stable number over the years. Historically this was done by dividing one's intelligence test score by one's chronological age. Researchers developed the idea of scaling intelligence test scores to produce a mean of 100 and a standard deviation of 15 (or 17, or 14, or …, depending on the people developing the test.)
Nowadays the scaling process is handled somewhat differently. An intelligence test is developed for a certain age range and group of people. The scores are then normed to produce a mean of 100 and a specified standard deviation such as 15 or some other number. Now, when a person in this age range takes this test, his or her IQ score is determined by looking up the test score in a table of values that converts test score to IQ by comparison with test scores of those used in the norming process.
Thus, for example, a person whose test score is close to the mean of the test scores used in the norming process will be assigned an IQ of approximately 100. Suppose that this person takes another IQ test ten years later. It may well be a different test with different questions, and designed to be suited to the person's current age. The score that he or she receives on the test will be compared to the scores achieved by people who were approximately this age when the test was created and normed. If his or her test score is near the mean of this norming group, the result will be an IQ of approximately 100.
Even with this norming process, IQ can change over time. As an example, consider a four year old who has grown up in extreme poverty and in a home and neighborhood environment that includes lead paint and other toxins, and a single parent who is holding down two jobs to make ends meet. Then the child's home environment changes markedly. Perhaps the single parent marries into greater wealth and the child now experiences a much better home and neighborhood environment. Moreover, the child goes to a high quality kindergarten and then on into high quality elementary school. Such changes can produce a marked increase in IQ.
Nature and Nurture
It is important to understand that intelligence depends on a combination of nature and nurture. On average, intelligence increases considerably as a person grows up, and it decreases as one grows old. It is the norming process in IQ that (artificially) makes it appear that one's intelligence is not changing over the years.
Here is a somewhat different way of looking at this question. A newborn with a healthy brain has a tremendous capacity to learn. The child's brain grows rapidly and learns rapidly. Just imagine the challenge of gaining oral fluency in one language. If the child happens to live in a bilingual or trilingual home and extended environment, the typical child will become bilingual or trilingual. Amazing! This represents a huge capacity to learn and to make use of one's learning.
My point is that the "average" person is very intelligent. Good informal and formal learning opportunities can greatly increase the Gc component of one's intelligence.
Studies of nature versus nurture are typically done making use of identical twins that were separated at birth. One can find varying results in the literature—with IQ being determined about 50 percent by nature all the way up to about 80 percent by nature, depending on the study.
Current research suggests that nature and nurture work together in a very complex manner, and that we have a long way to go in this area of research.
Multiple Intelligences
A human brain is a very complex organ. Many different parts and characteristics of a human brain contribute to its ability to learn and to deal with complex, challenging problems. Perhaps you have observed that some people seem to have more linguistic ability, or musical ability, or math ability that other people. Might there be significant discipline-oriented differences in intelligence? That is, might the differing characteristics of healthy brains have significant built-in inherent abilities to learn some disciplines better than others?
This question has led to a variety of multiple intelligences models and ongoing disagreements among proponents of these various models and proponents of a single factor model of intelligence. These are importatn disagreements. Suppose that nature can endow one person with the brain and hearing system to gain/have perfect pitch, another person to have much better than average propensity to develop very good spatial sense, and a third to have a propensity for developing much above average mathematical logic sense? If so, then perhaps educators would want to identify these varying characteristics in their students and better individualize programs of study to meet there varying students.
Howard Gardner is well known for his theory of Multiple Intelligences. His current theory includes eight different components, including logical/mathematical, and spatial. (Spatial intelligence is quite helpful in math as well as in other disciplines, such as art.)
Suppose that Howard Gardner's theory of multiple intelligences is essentially correct. Moreover, suppose that there are huge differences in the inherent and potential logical/mathematical abilities of students. If this is the case, then perhaps we are expecting many students in school to attempt to learn far more math than their brains are well suited to learn. Our school system's heavy emphasis on math puts many students at risk of not graduating from high school.
Personal comment: I am not tone deaf. However, my children like to label me as "tune" deaf. I tend to believe that I have somewhat below average inherent ability in music. It is not clear to me that I would have made it through high school if I had been required to take a stringent sequence of musical performance, composition, and theory courses in high school.
Howard Gardner based his theory of Multiple Intelligences partly on studies he did on brain damaged people. If ability in a particular area—such as spatial sense—is wiped out by damage to a particular part of one's brain, then this is taken as evidence that spatial is a distinct type of intelligence.
However, many people disagree with the work of Howard Gardner. One of the difficulties is that the various parts of one's brain work together, and that there is considerable plasticity.
To take a personal example, I (Dave Moursund) was quite good at math from my early childhood on, and I had little trouble in getting a doctorate in this discipline. It certainly helped that both my mother and father were high IQ people, and both were mathematicians. However, my spatial sense is terrible—well below average. At one time it was believe that a high spatial "IQ" was an essential part of being successful in math. More recent research indicates that many successful mathematicians have poor spatial sense. There is more than one way to look at a math problem!
Cognitive Development
Cognitive development is measured and studied in terms of a stage theory. Piaget is well known for the initial four-level stage theory that he developed. According to Piaget, a child moves from the Sensory Motor Stage to the Pre Operational Stage to the Concrete Operations Stage to the Formal Operations Stage.
The following chart is from the Piaget reference given above:
Quoting from the same reference:
Data from adolescent populations indicates only 30 to 35% of high school seniors attain the cognitive development stage of formal operations (Kuhn, Langer, Kohlberg & Haan, 1977). For formal operations, it appears that maturation establishes the basis, but a special environment is required for most adolescents and adults to attain this stage.
More modern versions of this stage theory have a much larger number of stages. See:
This article provides a 15-stage Piagetian-type model of cognitive development. Quoting from the article:
The acquisition of a new-stage behavior has been an important aspect of Piaget's theory of stage and stage change. Because of his controversial notions of stage and stage change, however, little research on these issues has taken place in the late twentieth century, at least among psychologists in the United States. The research that has taken place is being done by Neo-Piagetians. The neo-Piagetians more precisely defined stage, taking each of Piaget's substages and showing that they were in fact stages. In addition, three postformal stages have been added. Similar changes were made with Kohlberg's stages and substages. Commons, Richards, and Armon (1984) created a stage comparison table, comparing stage sequences from a number of different traditions, that stands today as the standard. This table shows that there is, essentially, only one stage sequence. Commons and Richards (1984a, b) presented their first General Stage Model at that time. Commons and colleagues (Commons, Trudeau, et al., 1998; Commons & Miller, 1998) later revised that model and expanded it downward, changing the name of the model to the Model of Hierarchical Complexity. Table 1a shows a complete list of the Orders of Hierarchical Complexity described in that model.
Researchers in cognitive development are faced by many of the same issues as researchers in IQ. Two of these issues are:
What (relative) roles do nature and nurture play in cognitive development?
Is cognitive development essentially domain independent or is does a theory of "multiple" cognitive developments better describe the field?
IQ and Cognitive Development are relatively closely related areas. (An IQ test and a cognitive development test may well make use of some of the same questions or activities.) A brain undergoes cognitive development from the time it first begins to form in a fetus. This cognitive development continues throughout life. (However, there can be cognitive decline due to age-related and other damage to the brain.)
IQ testing is one way to measure cognitive development. A test based on a stage theory of cognitive development is another approach to such measurement. Each of these measurement processes can be used to produce a number, such as an IQ number or a stage number.
The norming process in IQ measurement tends to produce a number that remains relatively stable over time. The stage measurement approach produces a stage level (number or designation) that increases over time as a person moves up a cognitive development scale.Relatively few people reach the top level of this 15-point scale.
Math Cognitive Development
There are a large number of hits on a Google search of math cognitive development. As an example, Stages of Math Development is a very short article that includes a math cognitive development stage theory model for children up to six years old.
Piaget did a lot of research in developing his 4-stage model of cognitive development. Besides his general interests in cognitive development, he also has a particular interest in math cognitive development. The following Dina and Pierre van Hiele geometry cognitive development scale was certainly inspired by Piaget's work. See also
Level
Name
Description
0
Visualization
Students recognize figures as total entities (triangles, squares), but do not recognize properties of these figures (right angles in a square).
1
Analysis
Students analyze component parts of the figures (opposite angles of parallelograms are congruent), but interrelationships between figures and properties cannot be explained.
2
Informal Deduction
Students can establish interrelationships of properties within figures (in a quadrilateral, opposite sides being parallel necessitates opposite angles being congruent) and among figures (a square is a rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.
3
Deduction
At this level the significance of deduction as a way of establishing geometric theory within an axiom system is understood. The interrelationship and role of undefined terms, axioms, definitions, theorems, and formal proof is seen. The possibility of developing a proof in more than one way is seen. (Roughly corresponds to Formal Operations on the Piagetian Scale.)
4
Rigor
Students at this level can compare different axiom systems (non-Euclidean geometry can be studied). Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.
Notice that the van Hieles, being mathematicians, labeled their first stage Level 0. This is a common practice that mathematicians use when labeling the terms of a sequence. Piaget's cognitive development scale has four levels, numbers 1 to 4. The highest level in the van Hiele geometry cognitive development scale is one level above the highest level of the Piaget cognitive development scale.
The following scale was created (sort of from whole fabric) by
David Moursund. It represents his current insights into a six-level, Piagetian-type, math cognitive development scale.
Stage & Name
Math Cognitive Developments
Level 1. Piagetian and Math sensorimotor. Birth to age 2.
Infants use sensory and motor capabilities to explore and gain increasing understanding of their environments. Research on very young infants suggests some innate ability to deal with small quantities such as 1, 2, and 3. As infants gain crawling or walking mobility, they can display innate spatial sense. For example, they can move to a target along a path requiring moving around obstacles, and can find their way back to a parent after having taken a turn into a room where they can no longer see the parent.
Level 2. Piagetian and Math preoperational. Age 2 to 7.
During the preoperational stage, children begin to use symbols, such as speech. They respond to objects and events according to how they appear to be. The children are making rapid progress in receptive and generative oral language. They accommodate to the language environments (including math as a language) they spend a lot of time in, so can easily become bilingual or trilingual in such environments.
During the preoperational stage, children learn some folk math and begin to develop an understanding of number line. They learn number words and to name the number of objects in a collection and how to count them, with the answer being the last number used in this counting process.
A majority of children discover or learn "counting on" and counting on from the larger quantity as a way to speed up counting of two or more sets of objects. Children gain increasing proficiency (speed, correctness, and understanding) in such counting activities.
In terms of nature and nurture in mathematical development, both are of considerable importance during the preoperational stage.
Level 3. Piagetian and Math concrete operations. Age 7 to 11.
During the concrete operations stage, children begin to think logically. In this stage, which is characterized by 7 types of conservation: number, length, liquid, mass, weight, area, volume, intelligence is demonstrated through logical and systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions that are reversible).
While concrete objects are an important aspect of learning during this stage, children also begin to learn from words, language, and pictures/video, learning about objects that are not concretely available to them.
For the average child, the time span of concrete operations is approximately the time span of elementary school (grades 1-5 or 1-6). During this time, learning math is somewhat linked to having previously developed some knowledge of math words (such as counting numbers) and concepts.
However, the level of abstraction in the written and oral math language quickly surpasses a student's previous math experience. That is, math learning tends to proceed in an environment in which the new content materials and ideas are not strongly rooted in verbal, concrete, mental images and understanding of somewhat similar ideas that have already been acquired.
There is a substantial difference between developing general ideas and understanding of conservation of number, length, liquid, mass, weight, area, and volume, and learning the mathematics that corresponds to this. These tend to be relatively deep and abstract topics, although they can be taught in very concrete manners.
Level 4. Piagetian and Math formal operations. After age 11.
Starting at age 11 or 12, or so, thought begins to be systematic and abstract. In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts, problem solving, and gaining and using higher-order knowledge and skills.
Math maturity supports the understanding of and proficiency in math at the level of a high school math curriculum. Beginnings of understanding of math-type arguments and proof.
Piagetian and Math formal operations includes being able to recognize math aspects of problem situations in both math and non-math disciplines, convert these aspects into math problems (math modeling), and solve the resulting math problems if they are within the range of the math that one has studied. Such transfer of learning is a core aspect of Level 4.
Level 4 cognitive development can continue well into college, and most students never fully achieve Level 4 math cognitive development. (This is because of some combination of innate math ability and not pursuing cognitively demanding higher level math courses or equivalent levels on their own.)
Mathematical content proficiency and maturity at the level of contemporary math texts used at the upper division undergraduate level in strong programs, or first year graduate level in less strong programs. Good ability to learn math through some combination of reading required texts and other math literature, listening to lectures, participating in class discussions, studying on your own, studying in groups, and so on. Solve relatively high level math problems posed by others (such as in the text books and course assignments). Pose and solve problems at the level of one's math reading skills and knowledge. Follow the logic and arguments in mathematical proofs. Fill in details of proofs when steps are left out in textbooks and other representations of such proofs.
Level 6. Mathematician.
A very high level of mathematical proficiency and maturity. This includes speed, accuracy, and understanding in reading the research literature, writing research literature, and in oral communication (speak, listen) of research-level mathematics. Pose and solve original math problems at the level of contemporary research frontiers.
Cognitive Acceleration In Mathematics Education
Shayer, Michael and Mundher, Adhami (June 2006). The Long-Term Effects from the Use of CAME (Cognitive Acceleration In Mathematics Education): Some Effects from the Use of the Same Principles in Y1&2, and the Maths Teaching of the Future. Proceedings of the British Society for Research into Learning Mathematics. Retrieved 8/14/09: Quoting from the article:
The CAME[1] project was inaugurated in 1993 as an intervention delivered in the context of mathematics with the intention of accelerating the cognitive development of students in the first two years of secondary education. This paper reports substantial post-test and long-term National examination effects of the intervention. The RCPCM project[2], an intervention for the first two years of Primary education, doubled the proportion of 7 year-olds at the mature concrete level to 40%, with a mean effect-size of 0.38 S.D. on Key Stage 1 Maths. Yet, instead of the intervention intention, it is now suggested that a better view is to regard CAME as a constructive criticism of normal instructional teaching, with implications for the role of mathematics teachers and university staff in future professional development.
BACKGROUND TO CAME AND RCPCM
In the mid-70s CSMS[3] survey 14,000 children aged 10 to 16 were given three Piagetian tests to assess the range of thinking levels at each year. Figure 2 shows the findings.
By [age] 14 only 20% were showing formal operational thinking (3A&3B). This mattered at the time because current O-level science and maths courses, designed for grammar-school children in the top 20% of the ability range, required this level from the end of Y8. In the 80s the Graded Assessment in Maths scheme for the ILEA found that by the age of 12 the children's mathematics competence had a 12-year developmental gap between the above-average and those at what would later be National Curriculum Levels 1 and 2.
The CASE project shows the need to teach math in a manner that helps to increase cognitive development. Quoting from the article:
Enough evidence has been presented to show that the research has engendered class management skills in the teachers involved that realise Vygotsky's insistence that teaching should foster development as well as subject knowledge: that it should always aim ahead of where students presently are.
David Tall
David Tall is a Professor of Mathematical Thinking at the University of Warwick. Quoting from the linked site:
My interests in cognitive development in mathematics have matured over the years. Initially, as a mathematics lecturer at university, seeing mathematics through the eyes of a mathematician, I used mathematical theories (such as catastrophe theory) to formulate cognitive theory. After researching concepts of limits and infinity, I designed computer software to visualize calculus ideas using the idea of 'local straightness' rather than formal limits, and expanded my interest in visualization. Working first with Michael Thomas on algebra, then Eddie Gray on arithmetic, I was able to see the links between symbolism in arithmetic, algebra and calculus. This led directly to the theory of procepts which is concerned essentially with symbols that represent both process and concept and the ability to switch flexibly between processes 'to do' and concepts 'to think about'. Eddie and I were then able to see what we termed 'the proceptual divide '- the bifurcation between those children who remain focused on more specific procedures rather than develop the flexibility of proceptual thinking.
As. As, click here.
Stage Theory in Math
In a 4/18/09 email message "Michael Lamport Commons" <commons@tiac.net> wrote concerning a 15 point scale with the stages numbered 0–14:
What we have found is that to pass courses such as linear algebra, multivariate calculus and the like requires systematic stage 11 reasoning, one stage beyond formal stage 10. To pass abstract mathematics classes like modern algebra, topology, etc, when they are proof based, require metaystematic stage 12 reasoning. We also find that this is the earliest domain in which such reasoning develops.
A detailed discussion of the 15-level stage theory is available in the following reference:
Abstract: The Model of Hierarchical Complexity presents a framework for scoring reasoning stages in any domain as well as in any cross cultural setting. The scoring is based not upon the content or the participant material, but instead on the mathematical complexity of hierarchical organization of information. The participant's performance on a task of a given complexity represents the stage of developmental complexity. This paper presents an elaboration of the concepts underlying the Model of Hierarchical Complexity (MHC), the description of the stages, steps involved in universal stage transition, as well as examples of several scoring samples using the MHC as a scoring aid.
Michael Commons' work in this area is important for two reasons:
1. It emphasizes that a stage theory can be developed that cuts across domains and culture.
The term "postformal" has come to refer to various stage characterizations of behavior that are more complex than those behaviors found in Piaget's last stage—formal operations—and generally seen only in adults. Commons and Richards (1984a, 1894b) and Fischer (1980), among others, posited that such behaviors follow a single sequence, no matter the domain of the task e.g., social, interpersonal, moral, political, scientific, and so on.
Most postformal research was originally directed towards an understanding of development in one domain. The common approach to much of the work on postformal stages has been to specify a performance on tasks that develop out of those described by Piaget (1950, 1952) as formal-operational or out of tasks in related domains (e.g., moral reasoning). The assumption has been made that the predecessor task performances (formal operations), are in some way necessary to the development of their successor performances and proclivities (postformal operations). Unlike many of the other theories, the Model of Hierarchical Complexity (MHC), presented here (Commons, et al., 1998), generates one sequence that addresses all tasks in all domains and is based on a contentless, axiomatic theory.
This document summarizes the research on identifying and describing four Piagetian-type cognitive development stages that are above Piaget's Stage 4: Formal Operations. The assertion is that these four stages apply equally well in every discipline. The quote from Commons given earlier in this section provides an example of examining these higher-level stages within the discipline of mathematics.
IQ and Stage Theory
Email from Michael Commons to David Moursund 5/10/09 says:
The MHC [Model of Hierarchical Complexity] shows that stages are absolute and do not need in any way norms. Hierarchical Complexity is a major determinant of how difficult a task is. So stage and IQ should be quite correlated. My guess, is about an r of .5.
…
The evidence for stage change is a lot more clearly studied than IQ change. Most intervention buy 1 or 2 stages at the most. I know of no studies showing more.
The first quoted part reemphasizes that IQ measures are normed and Cognitive Development measures on a Piagetian-type stage scale are not. Commons suggests that IQ and Stage level are moderately correlated.
As noted earlier in this document, the norming process used to measure IQ tends to make the measure of one's IQ remain fairly stable over time. However, there are interventions that can increase IQ. It is not clear whether there are long term studies that indicate such interventions lead to long term increase in IQ.
Finally, Common's last statement above and general research on stage theory indicate that intervention can increase the rate that people move through stages, and that interventions can move the top stage level reached up one or two steps on a 15-stage measure.
Math Learning Disorders
ScienceDaily (Oct. 27, 2005) — ROCHESTER, Minn. -- In—skills that are essential for success at school, work and for coping with life in general. The results appear in the September-October issue of Ambulatory Pediatrics.
…
In the current study, Mayo Clinic researchers used different definitions of Math LD, analyzed school records of boys and girls enrolled in public and private schools in Rochester, Minn., and examined information from the students' medical records. They also looked at the extent to which Math LD occurs as an isolated learning disorder versus the extent to which it occurs simultaneously with Reading LD. This study is the first to measure the incidence—the occurrence of new cases—of Math LD by applying consistent criteria to a specific population over a long time. By considering the coexistence of Math LD and Reading LD across the students' entire educational experience (i.e., from grades K-12), the research presents a more comprehensive description of this association.
Examples
What is dyscalculia? Dyscalculia is a term referring to a wide range of life-long learning disabilities involving math. There is no single form of math disability, and difficulties vary from person to person and affect people differently in school and throughout life.
What are the effects of dyscalculia? Since disabilities involving math can be so different, the effects they have on a person's development can be just as different. For instance, a person who has trouble processing language will face different challenges in math than a person who has difficulty with visual - spatial relationships. Another person with trouble remembering facts and keeping a sequence of steps in order will have yet a different set of math-related challenges to overcome.
Here is some material quoted from an article by Jerome Schultz: The article is about central auditory processing disorders (CAPD) as it relates to learning math.
Let me take this opportunity to help our readers understand CAPD a bit better, since this condition often goes unrecognized or is misdiagnosed as ADHD. The American Speech-Language-Hearing Association (ASHA) established a task force in 1996 to gain a better understanding of central auditory processing disorders (CAPD) in children.
…
Learning multiplication tables involves auditory pattern recognition, and temporal factors (the order of the language). Differentiating 8 x 7 = 56 from 6 x 7 = 42 is very difficult, since these are abstract symbols for a particular quantity. If she just says them over and over again, she may remember one...until she hears the next one. Your daughter has to be instructed in a concrete visual, hands-on way to understand ("see" in her mind's eye) that a number represents a quantity. Otherwise, the times tables are just another jumble of numbers.
David Geary
David Geary is a highly prolific researcher in math cognitive development.
Here are some David Geary papers available on the Web.
David C. Geary, Mary K. Hoard, Jennifer Byrd-Craven, Lara Nugent, and Chattavee Numtee (July/August 2007). Cognitive Mechanisms Underlying Achievement Deficits in Children With Mathematical Learning Disability. Child Development. Retrieved 4/9/09. To access this paper go to find the paper in the list of papers, and click on its link. Quoting from this paper:
Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n=15) and low achieving (LA, n=44) were identified. These groups and a group of typically achieving (TA, n=46) children were administered a battery of mathematical cognition, working memory, and speed of processing measures (M=6 years). The children with MLD showed deficits across all math cognition tasks, many of which were partially or fully mediated by working memory or speed of processing. Compared with the TA group, the LA children were less fluent in processing numerical information and knew fewer addition facts. Implications for defining MLD and identifying underlying cognitive deficits are discussed.
When viewed from the lens of evolution and human cultural history, it is not a coincidence that public schools are a recent phenomenon and emerge only in societies in which technological, scientific, commercial (e.g., banking, interest) and other evolutionarily-novel advances influence one's ability to function in the society (Geary, 2002, 2007). From this perspective, one goal of academic learning is to acquire knowledge that is important for social or occupational functioning in the culture in which schools are situated, and learning disabilities (LD) represent impediments to the learning of one or several aspects of this culturally-important knowledge. It terms of understanding the brain and cognitive systems that support academic learning and contribute to learning disabilities, evolutionary and historical perspectives may not be necessary, but may nonetheless provide a means to approach these issues from different levels of analysis. I illustrate this approach for MLD. I begin in the first section with an organizing frame for approaching the task of decomposing the relation between evolved brain and cognitive systems and school-based learning and learning disability (LD). In the second section, I present a distinction between potentially evolved biologically-primary cognitive abilities and biologically-secondary abilities that emerge largely as a result of schooling (Geary, 1995), including an overview of primary mathematics. In the third section, I outline some of the cognitive and brain mechanisms that may be involved in modifying primary systems to create secondary abilities, and in the fourth section I provide examples of potential the sources of MLD based on the framework presented in the first section.
Geary, David (March 2006). Dyscalculia at an Early Age: Characteristics and Potential Influence on Socio-Emotional Development. Encyclopedia on Early Childhood Development. Retrieved 4/9/09: Quoting from the first part of this paper:
Introduction. Dyscalculia refers to a persistent difficulty in the learning or understanding of number concepts (e.g. 4 > 5), counting principles (e.g. cardinality – that the last word tag, such as "four," stands for the number of counted objects), or arithmetic (e.g. remembering that 2 + 3 = "5"). These difficulties are often called a mathematical disability. We cannot yet predict which preschool children will go on to have dyscalculia, but studies that will allow us to develop early screening measures are in progress. At this time and on the basis of normal development during the preschool years, it is likely that preschoolers who do not know basic number names, quantities associated with small numbers (< 4), how to count small sets of objects, or do not understand that subtraction results in less and addition results in more are at risk for dyscalculia.
Subject: How Common is Dyscalculia? Between 3 and 8% of school-aged children show persistent grade-to-grade difficulties in learning some aspects of number concepts, counting, arithmetic, or in related math areas. These and other studies indicate that these learning disabilities, or dyscalculia, are not related to intelligence, motivation or other factors that might influence learning. The finding that 3 to 8% of children have dyscalculia is misleading in some respects. This is because most of these children have specific deficits in one or a few areas, but often perform at grade level or better in other areas. About half of these children are also delayed in learning to read or have a reading disability, and many have attention deficit disorder.
There have only been a few large-scale studies of children with MD [Mathematical Disability] and all of these have focused on basic number and arithmetic skills. As a result, very little is known about the frequency of learning disabilities in other areas of mathematics, such as algebra and geometry. In any case, the studies in number and arithmetic are very consistent in their findings: Between 6 and 7% of school-age children show persistent, grade-to-grade, difficulties in learning some aspects of arithmetic or related areas (described below). These and other studies indicate that these learning disabilities are not related to IQ, motivation or other factors that might influence learning.
The finding that about 7% of children have some form of MD is misleading in some respects. This is because most of these children have specific deficits in one or a few subdomains of arithmetic or related areas (e.g., counting) and perform at grade-level or better in other areas of arithmetic and mathematics. The confusion results from the fact that standardized math achievement tests include many different types of items, such as number identification, counting, arithmetic, time telling, geometry, and so fourth. Because performance is averaged over many different types of items, some of which children with MD have difficulty on and some of which they do not, many of these children have standardized achievement test scores above the 7th percentile (though often below the 20th).
Math Maturity
The previous parts of this document have explored Math IQ and Math Cognitive Development. They provide a theoretical underpinning for the discussion of math maturity given in the remainder of this document.
Components of Math Maturity
The term math maturity is widely used by mathematicians and math educators. For example, a middle school teacher may say, "I don't think Pat has the necessary math maturity to take an algebra course right now." It is clear that the teacher is not talking about Pat's math content knowledge.
Probably Pat has completed the prerequisite coursework. Perhaps Pat is weak in math reasoning and thinking, tends to learn math by rote memorization, has little interest in math, and shows little persistence in working on challenging math problems. The teacher feels that with this background, Pat is apt to struggle in algebra and likely fail the course.
At the university level, the dominant component in the literature of math maturity is "proof" and the logical,critical, creative reasoning and thinking involved in understanding and doing proofs. A person with a high level of math maturity has studied math at a level that requires substantial understanding of proof and regular demonstration of the ability to do proofs.
K-12 math has only a modest emphasis on formal proofs. However, as students move up in the math curriculum, they face a growing challenge to make mathematical arguments that describe and justify the steps they take in solving challenging math problems. This is a type of logical/mathematical (proof) activity.
The following list contains some components of math maturity. An increasing level of math maturity is demonstrated by:
1. An increasing capacity in the logical, critical, creative reasoning and thinking involved in understanding and solving problems and in understanding and doing proofs.
2. An increasing capacity to move beyond rote memorization in recognizing, posing, representing, and solving math problems. This includes transfer of learning of one's math knowledge and skills to problems in many different disciplines.
3. An increasing capability to communicate effectively in the language and ideas of mathematics. This includes:
A. Mathematical speaking and listening fluency.
B. Mathematical reading and writing fluency.
C. Thinking and reasoning in the language and images of mathematics.
4. An increasing capacity to learn mathematics—to build upon one's current mathematical knowledge and to take increasing personal responsibility for this learning.
5. Improvements in other factors affecting math maturity such as attitude, interest, intrinsic motivation, focused attention, perseverance and delayed gratification, having math-oriented habits of mind, and acceptance of and fitting into the "culture" of the discipline of mathematics.
Email from Joseph Dalin 6/14/09
The following email message sent to David Moursund was in response to a email message about math and Talented & Gifted education sent to an National Council of Supervisors of Mathematics distribution list.
Hi,
The major question is: what is a Talented and Gifted persons or students? What is his/her special capabilities? Memorizing or the capability of understanding? Do the Talented and Gifted students learn differently? Do they understand symbolic abstract language better, or significantly better, than ordinary students?
The basic education of mathematical understanding and creative thinking is gained through solving problems of cases which deal within the child's environment, experience and conceptual system.
Algebra is a symbolic abstract nonhuman language. In order to understand such a language there is a need to:
Have a "mathematical thinking maturity" and than—to translate the symbolic representations into graphic representation and learn through self experience, exploration and discovery (which is the way human beings learn).
It can be achieved by comprehensive integration of Visual-dynamic-quantitative computer software into the teaching and learning process of school mathematics.
Such an approach should be applied for all students, not only Talented and Gifted students.
I don't believe that, in general, students of 5th grade have the "mathematical learning maturity" for learning algebra 1.
I don't believe that most students of 7th grade are capable to leaning algebra 1 through its symbolic representation only. Anyhow, what's the rush? Learning is a long journey….
I don't believe that most students, even in higher grades, are capable to really understand Algebra by learning only through its symbolic representations. That's the main reason of poor achievement, failure and frustration of school mathematics education which is based on teaching symbolic mathematics.
Three of the paragraphs have been bolded for extra emphasis. The message is that math maturity is a key issue in determining when a student (whether TAG or not) should begin an algebra course. The message also suggests that current widely used methods for teaching introductory algebra do not adequately address the challenge of learning to deal with a high level of abstractness and translating it into personally meaningful understanding.
Clyde Greeno
The following is quoted from an email message sent by Clyde Greeno to the National Council of Supervisors of Mathematics distribution list on 4/9/09:
The entrenched "developmental" algebra curriculum (like the HS algebra curriculum) is a direct decedent from SMSG's calculus-preparatory HS algebra—which has served more to filter students out of the mathematics curriculum than to empower them for success within it. [No wonder that educators now are concerned about the "Algebra 2 for everyone" movement among state legislatures.]
Extensive clinical research has revealed that the primary cause for students' difficulties with algebra is simply that algebra curricula within the SMSG lineage badly violate scientifically established principles of the developmental psychology of mathematical learning—i.e. of "mathematics as common sense". [The original SMSG version was created two decades before America began to understand Piaget.] Ironically, the resulting "developmental" algebra curriculum is anything but developmental. The coming reformation will be guided by psychology.
Clinical methods quickly reveal that students learn the usual essentials of algebra better, faster, and more easily through the context of functions. That is partly because the field of algebra really is all about the study of operations/functions—even the SMSG curriculum was covertly about functions—even though that context still is badly hidden by the current curriculum.
Education for Increasing Math Maturity
This section is a work in progress.
Math maturity increases over time through:
General overall increase in cognitive development.
Learning math in a manner that facilitates higher-order creative thinking, problem solving, theorem proving, communicating in and about math, and learning to learn math.
Working with math teachers who have a higher level of math maturity than oneself, and being taught at a level that is a little above one's current level of math maturity.
Math maturity is strongly affected by one's informal education, formal education, and life experiences. As one's brain grows and as one is engaged in informal and formal education, one's overall intelligence grows and one's level of cognitive development grows. If one's education and experiences have an appropriate math component, one's math maturity will increase.
Assessment of Math Maturity
This section is a Work in Progress and definitely needs input from a lot of people.
It is relatively easy to make use of the term math maturity and to claim it is an important concept or goal in math education. It is much more difficult to develop assessment instruments that can be used for self-assessment (by students), for assessment by people interested in measuring how well our math education system is doing in helping students to develop math maturity, and as an aid to student placement in courses.
Good math teachers are able to estimate the math maturity of their students through a one-on-one conversation, by listening to the breadth and depth of questions a student raises in class, by listening to the breadth and depth of answers a student gives to questions raised during a class, through analysis of a student's homework and test answers, and so on. There are many clues available in these information sources. However, it is a major challenge to identify them and teach less qualified teachers to learn to observe and make use of these information sources.
Good math teachers can determine if a student has the math maturity to effectively deal with the content the teacher wants to teach and whether a student is apt to be bored by the level and pace of what is to be taught.
With some practice, students can gain skill in self-assessing their level of math maturity and progress they are making in increasing their level of math maturity. Part of a useful approach is a self-assessment based on insights into learning by rote memory versus learning for understanding. Another is self-assessment on dealing with "challenging" problems that draw upon math covered a few weeks ago, much earlier in the school year, and in previous school years.
Still another approach is through self-assessment of how well one can explain to oneself and to others the thought processes and understanding used in attacking challenging problems and proofs. In this, however, one needs to be aware that people who are good at math often have intuitive or not readily explained leaps of insight. Such leaps often do not lend themselves to the "show your work" type of requirement that most teachers require of their students.
Computers and Math Maturity
"Computers are incredibly fast, accurate, and stupid. Human beings are incredibly slow, inaccurate, and brilliant. Together they are powerful beyond imagination." (Albert Einstein)
"My familiarity with various software programs is part of my intelligence if I have access to those tools." (David Perkins,1992.)
The two quotes capture the essence of this section. An intact human mind and body has tremendous capabilities. However, it also has severe limitations. Over many thousands of years humans have been developing tools that help to overcome some of these physical and mental limitations.
Thus, for example, we have developed telescopes for "far seeing" and microscopes for "near seeing" that far exceed the capabilities of the human visual system. We have developed reading, writing, and arithmetic that are wonderful aids to one's brain. We have developed machines such as cars, airplanes, and bulldozers. We have developed highly automated manufacturing facilities.
Now, we have Information and Communication Technology. It plays a role in many of our previously developed tools, and it provides a new type of intelligence. Machine intelligence (artificial intelligence) can be thought of as a new type of brain, or as an auxiliary brain.
In terms of the document you are now reading, the major question is the nature and extent to which the computer brain adds to the capabilities of human intelligence and human cognitive development. That is, as educators we now need to think in terms of nature, nurture, and machine intelligence.
Here is a concrete example. Spatial intelligence in one of the eight Multiple Intelligences on Howard Gardner's list. We have long had maps and compasses to help people deal with certain types of spatial problems. We now have computerized maps (for example, think in terms of Google Earth) and GPS systems that can aid in solving some of the spatial problems that people face. In essence, such tools increase the intelligence of their users.
David Tall's Work
The article Tall (2000) discusses the cognitive load that is inherent to learning and using mathematics. The cognitive load is reduced through learning the language and symbolism of math so that one can use it rapidly and accurately at a subconscious level—in the same way that one uses their native language in speaking and writing.
Calculators and computers can play a significant role in math education. Quoting from Tall (2000):
The development of symbol sense throughout the curriculum
faces a number of major re-constructions causing increasing
difficulties to more and more students as they are faced with
successive new ideas that require new coping mechanisms.
For many it leads to the satisfying immediate short-term needs
of passing examinations by rote-learning procedures. The
students may therefore satisfy the requirements of the current
course and the teacher of the course is seen to be successful.
If the long-term development of rich cognitive units is not set in
motion, short-term success may only lead to increasing
cognitive load and potential long-term failure.
…
Given the constraints and support in the biological brain, the concept imagery in the mathematical mind can be very different from the working of the computational computer. A professional mathematician with mathematical cognitive units may use the computer in a very different way from the student who is meeting new ideas in a computer context.
The article then goes on to explain some of Tall's insights into what/how the brain is learning math in a graphing calculator or computer environment versus in the traditional paper and pencil environment. One way to think of this is in terms of the automation of tasks. A mathematician's brain has automated many tasks, and this has come through a considerable amount of practice. An alternative to this mental automation is, in a number of cases, to learn to use a graphing calculator or computer.
Thus, a math student who is a heavy user of graphing calculators and computers will be developing a type of math maturity that is different than that being developed by a person who does not become proficient in the use of these math tools. Tall does not argue that one type of approach to math education is superior to the other—just that certain aspects of the final results in a student's math-brain will be different. (See also: Moursund, 1986, 1988.)
Automaticity
This section is a work in progress.
Research into how people solve problems and gain in expertise within a particular problem-solving domain have helped us to understand how study and practice lead to increased automaticity and less demands on one's brain. Thus, for example, an expert chess player can recognize and process possible desirable moves in a complex board position much more rapidly and accurately than can an less qualified chess player. This speed of recognition and analysis has come from many thousands of hours of study and practice.
A similar type of learning occurs in math. Through thousands of hours of study and practice, a mathematician's brain automates a large number of problem recognition and possible action tasks. Thus, when faced by a new and challenging math problem, the expert mathematician is able to devote more brain power to the new and challenging parts while automaticity takes care of the familiar parts.
The chess and math examples apply to all areas in which a person can achieve a high level of expertise.
"As a task to be learned is practiced, its performance becomes more and more automatic; as this occurs, it fades from consciousness, the number of brain regions involved in the task becomes smaller." (A Universe Of Consciousness How Matter Becomes Imagination. Edelman & Tononi, 2000, p.51)
That is, through repetition of a task, a brain becomes more efficient at carrying out the task. But, it can take a lot of repetitions plus occasional practice to build and maintain this efficiency. An alternative in some cases is to just turn such repetitive task over to a computer or a calculator. Quoting from David Tall's 1996 article Can All Children Climb the Same Curriculum Ladder?:
This presentation presents evidence that the way the human brain thinks about mathematics requires an ability to use symbols to represent both process and concept. The more successful use symbols in a conceptual way to be able to manipulate them mentally. The less successful attempt to learn how to do the processes but fail to develop techniques for thinking about mathematics through conceiving of the symbols as flexible mathematical objects. Hence the more successful have a system which helps them increase the power of their mathematical thought, but the less successful increasingly learn isolated techniques which do not fit together in a meaningful way and may cause the learner to reach a plateau beyond which learning in a particular context becomes difficult.
The computer is quite different from the biological brain and therefore can be of value by providing an environment that complements human activity. Whilst the brain performs many activities simultaneously and is prone to error, the computer carries out individual algorithms accurately and with great speed. Computer calculations with numbers and manipulation of symbols has some similarities with the notion of procept. Internal computer symbolism is used both to represent data and also to perform routines to manipulate that data. However, there are significant differences. The computer is simply a device which manipulates information in a way specified by a program. It has none of the cognitive richness (or baggage) of the concept image available to the human mind to guide (or confuse) problem-solving activities.
This quote captures some of the idea of students learning by rote (sort of like a computer) and students learning with understanding as well with mental links to related topics that they know something about. This richer learning is a goal in math education. Here is a related quote from Tall:
The development of symbol sense throughout the curriculum therefore faces a number of major reconstructions which cause increasing difficulties to more and more students as they are faced with successive new ideas that require new coping mechanisms. For many it leads to the satisfying immediate short-term needs of passing examinations by rote-learning procedures. The students may therefore satisfy the requirements of the current course and the teacher of the course is seen to be successful. However, if the long-term development of rich cognitive units is not set in motion, short-term success may only lead to increasing cognitive load and potential long-term failure.
Learners have changed as a result of their exposure to technology, says Greenfield, who analyzed more than 50 studies on learning and technology, including research on multi-tasking and the use of computers, the Internet and video games. Her research was published this month in the journal Science.
Reading for pleasure, which has declined among young people in recent decades, enhances thinking and engages the imagination in a way that visual media such as video games and television do not, Greenfield said.
…
Visual intelligence has been rising globally for 50 years, Greenfield said. In 1942, people's visual performance, as measured by a visual intelligence test known as Raven's Progressive Matrices, went steadily down with age and declined substantially from age 25 to 65. By 1992, there was a much less significant age-related disparity in visual intelligence, Greenfield said.
"In a 1992 study, visual IQ stayed almost flat from age 25 to 65," she said.
Once Mischel began analyzing the results, he noticed that low delayers, the children who rang the bell quickly, seemed more likely to have behavioral problems, both in school and at home. They got lower S.A.T. scores. They struggled in stressful situations, often had trouble paying attention, and found it difficult to maintain friendships. The child who could wait fifteen minutes had an S.A.T. score that was, on average, two hundred and ten points higher than that of the kid who could wait only thirty seconds.
New research funded by the Economic and Social Research Council (ESRC) and conducted by Michael Shayer, professor of applied psychology at King's College, University of London, concludes that 11- and 12-year-old children in year 7 are "now on average between two and three years behind where they were 15 years ago", in terms of cognitive and conceptual development.
"It's a staggering result," admits Shayer, whose findings will be published next year in the British Journal of Educational Psychology. "Before the project started, I rather expected to find that children had improved developmentally. This would have been in line with the Flynn effect on intelligence tests, which shows that children's IQ levels improve at such a steady rate that the norm of 100 has to be recalibrated every 15 years or so. But the figures just don't lie. We had a sample of over 10,000 children and the results have been checked, rechecked and peer reviewed."
This book includes an emphasis on thinking about problem solving partly from the point of view of developing a repertoire of smaller problems or problem-solving activities to a high level of automaticity or making use of comptuters as an substitute for part of this learning task.
Author or Authors
Math Maturity, Defined
Mathematicians tend to prefer the concept of math maturity over the idea of math cognitive development. A Google search (10/6/08) of the expression: "math maturity" OR "mathematical maturity" OR "mathematics maturity" produced over 24,000 hits. Wikipedia states:
Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught, but instead comes from repeated exposure to complex mathematical concepts.
Still quoting from the Wikipedia, other aspects of mathematical maturity include:
the capacity to generalize from a specific example to broad concept
the capacity to handle increasingly abstract ideas
the ability to communicate mathematically by learning standard notation and acceptable style
a significant shift from learning by memorization to learning through understanding
the capacity to separate the key ideas from the less significant
the ability to link a geometrical representation with an analytic representation
the ability to translate verbal problems into mathematical problems
the ability to recognize a valid proof and detect 'sloppy' thinking
the ability to recognize mathematical patterns
the ability to move back and forth between the geometrical (graph) and the analytical (equation)
improving mathematical intuition by abandoning naive assumptions and developing a more critical attitude
Thirty percent of mathematical maturity is fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas. Mathematics, like English, relies on a common understanding of definitions and meanings. But in mathematics definitions and meanings are much more often attached to symbols, not to words, although words are used as well. Furthermore, the definitions are much more precise and unambiguous, and are not nearly as susceptible to modification through usage. You will never see a mathematical discussion without the use of notation!
You can evaluate a math lesson plan or unit of study in terms of how it contributes to students gaining in math maturity.
The general notion of "maturity" in a discipline applies to every discipline—indeed to every job, vocation, or pastime. However, mathematics teachers have been engaged with the notion more often than teachers of other academic disciplines.
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"TI-83 Plus Graphing Calculator For Dummies" shows you how to: perform basic arithmetic operations; use Zoom and panning to get the best screen display; use all the functions in the Math menu, including the four submenus - MATH, NUM, CPS, and PRB; use the fantastic Finance application to decide whether to lease or get a loan and buy, calculate the best interest, and more; and, graph and analyze functions by tracing the graph or by creating a table of functional values, including graphing piecewise-defined and trigonometric functions.This title also shows you how to: explore and evaluate functions, including how to find the value, the zeros, the point of intersection of two functions, and more; draw on a graph, including line segments, circles, and functions, write text on a graph, and do freehand drawing; work with sequences, parametric equations, and polar equations; use the Math Probability menu to evaluate permutations and combinations; enter statistical data and graph it as a scatter plot, histogram, or box plot, calculate the median and quartiles, and more; and, deal with matrices, including finding the inverse, transpose, and determinant and using matrices to solve a system of linear equations. ...
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Out of stock. This title will be placed on back order and we will advise when it is shipped.
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Lecture 24: Introduction to Ratios
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Lecture Details :
Basic ratio problems. (from KhanAcademy.org)
Course Description :
This is the original Algebra course on the Khan Academy and is where Sal continues to add videos that are not done for some other organization. It starts from very basic algebra and works its way through algebra II.
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