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In worksheet 1.2 we talked about factors of whole numbers. Remember, if ab = ab then a is a factor of ab and b is a factor of ab. In a similar way we can look at the factors of an algebraic expression. So, for instance, 3uv has factors 1; 3; u; v and combinations of these like 3u; 3v; uv and of course 3uv.

The highest common factor is, as was the case with numbers, the biggest or largest factor that divides two expressions. So the highest common factor of 3uv and 6u (from example 1(a)) is 3u; the highest common factor of 2xy and 4xyz (from example 1(b)) is 2xy. As with whole numbers we can also nd the smallest algebraic expression that is a multiple of two expressions. This is called the lowest common multiple. Download free Factors of Algebraic Expressions.pdf here

An expression such as 5x 7x has two terms. These terms are called like terms because they have the same variable. You can use the Distributive Property to simplify expressions that have like terms. An expression is in its simplest form when it has no like terms and no parentheses. Download free Simplifying Algebraic Expressions.pdf here

This paper describes a summary of an experimental study that I have conducted at the University of Wisconsin -Whitewater in the Fall of 1994. I have taught three out of twenty-four Sections of Intermediate Algebra using Cooperative Learning approach together with Mentors. The goal was to investigate the impact of this approach on students’ performance by comparing the success rate of each Section (in the Pilot Group) against 21 Sections in Control Group. The study has shown that the students in two Pilot Sections made a significant improvement in their test scores.

Since 1987 I have been conducting experimental studies in teaching different undergraduate mathematics courses using Cooperative Learning approach. The results were positive and the teaching style, indeed, enhanced students’ learning and performance. In Fall 1994, with the strong support of campus administration I have decided to implement this approach together with mentors in teaching three out of twenty-four Sections of Intermediate Algebra. This project was funded by a grant from the University of Wisconsin- Whitewater. The fund has been used to hire four students(to serve as Mentors and grade weekly homework assignments), and also to pay a part-time academic staff to grade all exams for Pilot Group. Download free Teaching Intermediate Algebra using Cooperative Learning and Mentor.pdf here

There cannot be an exact answer to this question, unless the curriculum at all of the colleges in Maryland is standardized. Since this is not a practical, nor a desirable objective, a less precise definition will have to suffice. First of all, to answer this question, it was felt that a comparison between Introductory Algebra and Intermediate Algebra is necessary. Secondly, it is recognized that Intermediate Algebra is ideally a level of mathematical maturity, that can be achieved though a variety of topics and skills. Finally, Intermediate Algebra is not static; it is changing and evolving along with the technology available to teach mathematics. Although we will not give a precise definition of Intermediate Algebra here, we hope that once you have perused this document, you will be able to recognize Intermediate Algebra when you see it.

It is felt that a comparison of Introductory Algebra and Intermediate Algebra is necessary, because some topics found in Intermediate Algebra at one college may be found in Introductory Algebra at another college. Intermediate Algebra comes at the end of the sequence of developmental math courses. It is this sequence of courses that is, in reality, the true prerequisite to “College Level” mathematics courses; therefore, it is not important in which developmental course individual topics appear. Intermediate Algebra should be thought of as a level of mathematical maturity. It is a combination of computational skills, manipulative skills and critical thinking skills needed to “think” mathematically. These skills can be achieved through a variety of topics and pedagogical techniques. Download free What is Intermediate Algebra?.pdf here

If it is known that (1) some statement is true for n = 1 (2) assumption that statement is true for n implies that the statement is true for (n + 1) then the statement is true for all positive integers Modifications of the Principle of Mathematical Induction • If it is known that (1) some statement is true for n = n0 (positive integer) (2) assumption that statement is true for n implies that the statement is true for (n + 1) then the statement is true for all positive integers greater or equal to n0 • If it is known that (1) some statement is true for n = 1 (2) assumption that statement is true for all positive integers k, 1 k n implies that the statement is true for (n + 1) then the statement is true for all positive integers • (Backward induction) If it is known that (1) some statement is true for n = 1 (2) assumption that statement is true for n > 1 implies that the statement is true for 2n and (n − 1) then the statement is true for all positive integers

Mathematical Induction in Algebra 1. Prove that any positive integer n > 1 is either a prime or can be represented as product of primes factors. 2. Set S contains all positive integers from 1 to 2n. Prove that among any n + 1 numbers chosen from S there are two numbers such that one is a factor of the other. 3. Prove that if (x+1/x) is integer then (xn+1/xn) is also integer for any positive integer n. 4. For sequence of Fibonacci numbers u1 = 1, u2 = 1, uk+1 = uk+uk−1, k = 2, 3, . . . prove the formula uk+m = uk−1um + ukum+1 5. Prove the following identities: (a) 12 + 22 + 32 + · · · + n2 = n(n + 1)(2n + 1)/6 (b) 13 + 23 + 33 + · · · + n3 = n2(n + 1)2/4 (c) 1 × 2 × 3 + 2 × 3 × 4 + · · · + n(n + 1)(n + 2) = n(n + 1)(n + 2)(n + 3)/4 (d) 1 × 1! + 2 × 2! + · · · + n × n! = (n + 1)! − 1 6. Prove the following divisibilities: (a) 6/n3 + 5n (b) 7/(62n−1 + 1) (c) 3n+1/23n + 1 7. Prove the following inequalities: (a) 1/(n + 1) + 1/(n + 2) + 1/(n + 3) + · · · + 1/2n > 13/24 (n > 1) (b) 1/12 + 1/22 + 1/32 + · · · + 1/n2 < id="dwlinks" rel="nofollow" href="http://www.math.toronto.edu/oz/turgor/Induction.pdf" target="_blank">here

Mathematical induction is a beautiful tool by which one is able to prove in nitely many things with a nite amount of paper and ink. It works by exploiting underlying structure: a complex and unwieldy problem can sometimes be broken apart along its fault lines so as to leave behind many smaller problems, each of which is more easily solved. Induction is one method of nding such fault lines and organizing the smaller pieces of a larger problem. Often, the simpler structure of the small pieces permeates the whole, and a complicated structure can be seen to operate based on the same simple rules that govern its pieces. This can lead to results that are both powerful and counter-intuitive.

Induction is only one of many techniques through which one may attempt to wrestle with in nity in nite terms (which is to say: with home eld advantage), but it holds a rather distinguished position in mathematics. Conveniently, it requires very little background knowledge to learn, and for this reason it is often taught in high school and could reasonably be included in an elementary school curriculum. Its home is in the natural numbers : 1; 2; 3; 4; : : :, which are, barring geometrical objects, arguably the most intuitive of all mathematical objects. Despite its apparent simplicity, its use in contemporary mathematics is widespread. But perhaps most tellingly, a casual lunchtime conversation with my colleagues about induction revealed that everyone seemed to have their own \induction story", a tale of their rst encounter with or rst appreciation of mathematical induction. It is clear that induction holds a special place in the mathematician's heart, and so it is no surprise that it can be the source of so much beauty, confusion, and surprise. Download free Puzzles and Paradoxes in Mathematical Induction.pdf here

How does statistical thinking differ from mathematical thinking? What is the role of mathematics in statistics? If you purge statistics of its mathematical content, what intellectual substance remains? In what follows, we offer some answers to these questions and relate them to a sequence of examples that provide an overview of current statistical practice. Along the way, and especially toward the end, we point to some implications for the teaching of statistics.

Statistics is a methodological discipline. It exists not for itself but rather to offer to other fields of study a coherent set of ideas and tools for dealing with data. The need for such a discipline arises from the omnipresence of variability. Individuals vary. Repeated measurements on the same individual vary. In some circumstances, we want to find unusual individuals in an overwhelming mass of data. In others, the focus is on the variation of measurements. In yet others, we want to detect systematic effects against the background noise of individual variation. Statistics provides means for dealing with data that take into account the omnipresence of variability. Download free Mathematics, Statistics, and Teaching.pdf here

The College Board has developed standards for mathematics and statistics to help states, school districts, and schools provide all students with the rigorous education that will prepare them for success in college, opportunity in the workplace, and effective participation in civic life. The College Board’s commitment to this project is founded on the belief that all students can meet high expectations for academic performance when they are taught to high standards by qualified teachers.

College Board programs and services have supported the transition from high school to college for more than 100 years. Advanced Placement Program® (AP®) courses enable students to transition into college-level study when they are ready, even while still in high school. The SAT® Reasoning Test™, the SAT Subject Tests™, and the PSAT/NMSQT® all measure content knowledge and critical thinking and reasoning skills that are foundations for success in college. The College Board Standards for College Success makes explicit these college readiness skills so that states, school districts, and schools can better align their educational programs to clear definitions of college readiness.

Preparing students for college before they graduate from high school is critical to students’ completing a college degree. Most college students who take remedial courses fail to earn a bachelor’s degree (Adelman, 2004). To reduce the need for remediation in college, K–12 educational systems need clear and specific definitions of the knowledge and skills that students should develop by the time they graduate in order to be prepared for college success. By aligning curriculum, instruction, assessment, and professional development to clear definitions of college readiness, schools can help reduce the need for remediation in college and close achievement gaps among student groups, ultimately increasing the likelihood that students will complete a college degree. Download free Mathematics and Statistics College Board Standards for College Success.pdf here

This book is a course of lectures on the mathematics of actuarial science. The idea behind the lectures is as far as possible to deduce interesting material on contingent present values and life tables directly from calculus and common- sense notions, illustrated through word problems. Both the Interest Theory and Probability related to life tables are treated as wonderful concrete appli- cations of the calculus. The lectures require no background beyond a third semester of calculus, but the prerequisite calculus courses must have been solidly understood. It is a truism of pre-actuarial advising that students who have not done really well in and digested the calculus ought not to consider actuarial studies.

It is not assumed that the student has seen a formal introduction to prob- ability. Notions of relative frequency and average are introduced ¯rst with reference to the ensemble of a cohort life-table, the underlying formal random experiment being random selection from the cohort life-table population (or, in the context of probabilities and expectations for `lives aged x', from the subset of lx members of the population who survive to age x). The cal- culation of expectations of functions of a time-to-death random variables is rooted on the one hand in the concrete notion of life-table average, which is then approximated by suitable idealized failure densities and integrals. Later, in discussing Binomial random variables and the Law of Large Numbers, the combinatorial and probabilistic interpretation of binomial coe±cients are de- rived from the Binomial Theorem, which the student the is assumed to know as a topic in calculus (Taylor series identi¯cation of coe±cients of a poly- nomial.) The general notions of expectation and probability are introduced, but for example the Law of Large Numbers for binomial variables is treated (rigorously) as a topic involving calculus inequalities and summation of ¯nite series. This approach allows introduction of the numerically and conceptually useful large-deviation inequalities for binomial random variables to explain just how unlikely it is for binomial (e.g., life-table) counts to deviate much percentage-wise from expectations when the underlying population of trials is large. Download free Actuarial Mathematics and Life-Table Statistics.pdf here

We hope that a student, interested in mathematics as a major or because of its role in their career path, will ponder these questions guided by the brief re ections given here. Mathematics has been called the "Queen of the Sciences". History reveals to us the reasons for this and reveals the dual nature of mathematics providing guiding light toward both the above questions. On one hand, mathemat- ics is a content driven discipline constructed via pure mathematical research following the deductive paradigm of axioms, theorems and proofs. This be- gan primarily in the Greek Era over 2000 years ago with the development of Euclidean geometry and overtime has owered into deep areas of mathemat- ical thought: algebra, analysis, combinatorics, geometry, and number theory to name a few. On the other hand, mathematics is a powerful language to model and understand the physical world. Indeed, mathematics provides a tool in physics providing foundational models in mechanics, thermodynam- ics, electricity and magnetism, quantum mechanics, and relativity. In the 19th and 20th centuries, the breadth of mathematical applications grew im- mensely: the industrial age spawned applications of mathematics thoughout the enginneering world; applications in modeling population dynamics, cel- lular function and dynamics were discovered leading to a diversity of mathe- matical thought in the biological sciences and medicine; the computer age has led to new results in discrete mathematics and in applying number theory to encryption algorithms; and most recently mathematics has found signi cant application in the analysis of nancial markets. The meaning of the word "dual" should be clear: often, the results of mathematics provide the tool in the applied world and just as often, the desire to construct a model leads to new mathematical ideas on which to apply the deductive paradigm. Download free Undergraduate Handbook for Mathematics & Statistics.pdf here

This handbook is directed primarily to mathematics majors and minors; its purpose is to provide useful advice and information so that students can get the most out of their studies in mathematics. This handbook should also be a useful resource for potential majors and minors and for university personnel who advise students. The information and policies set forth here are intended to supplement material contained in the Bulletin of Duke University 2008–2009: Undergraduate Instruction.

The information in this handbook applies to the academic year 2008-2009 and is accurate and current, to the best of our knowledge, as of August 2008. Inasmuch as changes may be necessary from time to time, the information contained herein is not binding on Duke University or the Duke University Department of Mathematics, and should not be construed as constituting a contract between Duke University and any individual. The University reserves the right to change programs of study, academic requirements, personnel assignments, the announced University calendar, and other matters described in this handbook without prior notice, in accordance with established procedures. Download free Handbook for Mathematics Majors and Minors.pdf here