# Review of Algebra 2 REVIEW of ALGEBRA

Total Page：16

File Type：pdf, Size：1020Kb

Recommended publications
• LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related diﬃculty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoﬀ in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of ﬁnite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inﬂated, the reader is kindly invited to open the ﬁrst chapter of the book, read the ﬁrst page to the point where the ﬁrst result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The eﬀect would, of course, be magniﬁed if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
[Show full text]
• Schaum's Outline of Linear Algebra (4Th Edition)
[Show full text]
• Problems in Abstract Algebra
[Show full text]
• From Arithmetic to Algebra
From arithmetic to algebra Slightly edited version of a presentation at the University of Oregon, Eugene, OR February 20, 2009 H. Wu Why can’t our students achieve introductory algebra? This presentation speciﬁcally addresses only introductory alge- bra, which refers roughly to what is called Algebra I in the usual curriculum. Its main focus is on all students’ access to the truly basic part of algebra that an average citizen needs in the high- tech age. The content of the traditional Algebra II course is on the whole more technical and is designed for future STEM students. In place of Algebra II, future non-STEM would beneﬁt more from a mathematics-culture course devoted, for example, to an understanding of probability and data, recently solved famous problems in mathematics, and history of mathematics. At least three reasons for students’ failure: (A) Arithmetic is about computation of speciﬁc numbers. Algebra is about what is true in general for all numbers, all whole numbers, all integers, etc. Going from the speciﬁc to the general is a giant conceptual leap. Students are not prepared by our curriculum for this leap. (B) They don’t get the foundational skills needed for algebra. (C) They are taught incorrect mathematics in algebra classes. Garbage in, garbage out. These are not independent statements. They are inter-related. Consider (A) and (B): The K–3 school math curriculum is mainly exploratory, and will be ignored in this presentation for simplicity. Grades 5–7 directly prepare students for algebra. Will focus on these grades. Here, abstract mathematics appears in the form of fractions, geometry, and especially negative fractions.
[Show full text]
[Show full text]
• Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students
EDUCATOR’S PRACTICE GUIDE A set of recommendations to address challenges in classrooms and schools WHAT WORKS CLEARINGHOUSE™ Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students NCEE 2015-4010 U.S. DEPARTMENT OF EDUCATION About this practice guide The Institute of Education Sciences (IES) publishes practice guides in education to provide edu- cators with the best available evidence and expertise on current challenges in education. The What Works Clearinghouse (WWC) develops practice guides in conjunction with an expert panel, combining the panel’s expertise with the findings of existing rigorous research to produce spe- cific recommendations for addressing these challenges. The WWC and the panel rate the strength of the research evidence supporting each of their recommendations. See Appendix A for a full description of practice guides. The goal of this practice guide is to offer educators specific, evidence-based recommendations that address the challenges of teaching algebra to students in grades 6 through 12. This guide synthesizes the best available research and shares practices that are supported by evidence. It is intended to be practical and easy for teachers to use. The guide includes many examples in each recommendation to demonstrate the concepts discussed. Practice guides published by IES are available on the What Works Clearinghouse website at http://whatworks.ed.gov. How to use this guide This guide provides educators with instructional recommendations that can be implemented in conjunction with existing standards or curricula and does not recommend a particular curriculum. Teachers can use the guide when planning instruction to prepare students for future mathemat- ics and post-secondary success.
[Show full text]
• Taming the Unknown. a History of Algebra from Antiquity to the Early Twentieth Century, by Victor J
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 725–731 S 0273-0979(2015)01491-6 Article electronically published on March 23, 2015 Taming the unknown. A history of algebra from antiquity to the early twentieth century, by Victor J. Katz and Karen Hunger Parshall, Princeton University Press, Princeton and Oxford, 2014, xvi+485 pp., ISBN 978-0-691-14905-9, US \$49.50 1. Algebra now Algebra has been a part of mathematics since ancient times, but only in the 20th century did it come to play a crucial role in other areas of mathematics. Algebraic number theory, algebraic geometry, and algebraic topology now cast a big shadow on their parent disciplines of number theory, geometry, and topology. And algebraic topology gave birth to category theory, the algebraic view of mathematics that is now a dominant way of thinking in almost all areas. Even analysis, in some ways the antithesis of algebra, has been aﬀected. As long ago as 1882, algebra captured some important territory from analysis when Dedekind and Weber absorbed the Abel and Riemann–Roch theorems into their theory of algebraic function ﬁelds, now part of the foundations of algebraic geometry. Not all mathematicians are happy with these developments. In a speech in 2000, entitled Mathematics in the 20th Century, Michael Atiyah said: Algebra is the oﬀer made by the devil to the mathematician. The devil says “I will give you this powerful machine, and it will answer any question you like. All you need to do is give me your soul; give up geometry and you will have this marvellous machine.” Atiyah (2001), p.
[Show full text]
• Basic Algebra Review
SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 1 Algebra Review Exponents (continued) Polynomials (continued) Factoring (continued) Rational Expressions Rational Expressions Equations of Lines Quotient Rules FOIL Expansion for Multiplying Two To find the value(s) for which a rational (continued) TwoVariables (continued) Factoring Trinomials, Z Binomials expression is undefined, set the denominator If a 0, Leading Term Z x2 Intercepts 0 equal to 0 and solve the resulting equation. SIMPLIFYING COMPLEX FRACTIONS i. Zero exponent: a = 1 i. Multiply the first terms. 2 + + Z To find the x-intercept, let y = 0. To factor ax bx c, a 1: Lowest Terms Numbers Linear Equations ii. Multiply the outer terms. Method 1 To find the y-intercept, let x = 0. -n = 1 By Grouping To write a rational expression in lowest terms: ii. Negative exponents: a n iii. Multiply the inner terms. i. Simplify the numerator and denominator FRACTIONS Definition of Subtraction Properties a i. Find m and n such that i. Factor the numerator and denominator. Slope iv. Multiply the last terms. = + = separately. m mn ac and m n b. Suppose (x , y ) and (x , y ) are two differ- Addition and Subtraction x - y = x + -y i. Addition: The same quantity may be a - ii. Divide out common factors. ii. Divide by multiplying the simplified 1 1 2 2 1 2 iii. Quotient rule: = m n v. Collect like terms. Z added to (or subtracted from) each side of n a ii. Then numerator by the reciprocal of the ent points on a line. If x1 x2, then the i.
[Show full text]
• Linear Algebra Methods in Combinatorics
Linear Algebra methods in Combinatorics Dani¨elKroes Jacob Naranjo Jiaxi Nie [email protected] [email protected] [email protected] Jason O'Neill Nicholas Sieger Sam Sprio Emily Zhu [email protected] [email protected] [email protected] [email protected] Fall 2019 Quarter Abstract Linear algebra tools have been used to solve many problems in extremal combinatorics. The far reaching nature of the subject matter has resulted in a book [4] written by Babai and Frankl. Many of the proofs in the area are short, elegant and straightforward and as a result ﬁt perfectly in a graduate student seminar. We will start with a few results which are fundamental in extremal set theory including the oddtown/eventown problem and the Frankl-Wilson Theorem. The seminar will culminate with some big results in the past few years including the method of slice rank to solve capset problem and the recently solved sensitivity conjecture. 1 Introduction These notes are comprised from an eight lecture series for graduate students in combinatorics at UCSD during the Fall 2019 Quarter. The organization of these expository notes is as follows. Each section corresponds to a ﬁfty minute lecture given as part of the seminar. We shall ﬁrst establish some common notation. 1.1 Notation Throughout these notes, we let [n] := f1; 2; : : : ; ng and consider 2[n] := fA ⊂ [n]g: Moreover, we are interested in the collection of size k subsets of an n-element set which we denote [n] as k := fA ⊂ [n] : jAj = kg: [n] [n] We denote subfamilies of k and 2 by calligraphic letters, sets by capital letters and elements by lower case letters.
[Show full text]
• Elementary Algebra > Syllabus
San Antonio College · - · MATH-Mathematics Elementary Algebra MATH-0410 Summer - 8 Week Session Summer 2018 Section 026.14294 4-4-1 Credits 06/04/2018 to 07/26/2018 Modified 05/31/2018 Contact Information Professor Roland Treviño McCreless Hall, MCCH 126C [email protected] (Canvas email preferred) Math Department – MCCH 221 – 210.486.0270 Administrative Services Specialist - Patricia Gonzalez – [email protected] Academic Unit Assistant - Cynthia Morton – [email protected] Program Coordinator - Paula McKenna – [email protected] Department Chair – Dr. Said Fariabi – [email protected] Materials PREREQUISITES, CO-REQUISITES and OTHER REQUIREMENTS: Prerequisite(s):TSI score MATH Numeracy 310-335 with ABE 3 -6. Course placement advisement is available in the Mathematics/Computer Science Office located at MCCH 221. TEXTBOOKS (including ISBN#) and REQUIRED MATERIALS/RECOMMENDED READINGS: The student should go to www.connectmath.com to register for the online math program that accompanies this course. Register as a new student even if you have used this program in the past. You will be asked for the following Course Code: E4AFH-YHTVQ You already purchased access to this program at registration. There is no need for any more purchases beyond this. COURSE CONTENT: Topics include those listed below. Please see the Methods of Measurement section below to see other requirements such as exams. Chapter 1: Whole Numbers 1.2 – 1.6 (REVIEW ONLY) 1.7 Exponents, Algebraic Expressions, and the Order of Operations Chapter 2: Integers and Algebraic
[Show full text]
• Mathematics.Pdf
Algebra Algebra II w/ Trigonometry AP Calculus AB AP Calculus BC AP Statistics Calculus III College Algebra Geometry Linear Algebra Pre-Calculus Pre-Calculus Career Internship Program Mathematics 101 Mr. Collin Voigt, Division Chair TEL: SC: (708) 579-6580, NC: (708) 579-6410 FAX: (708) 579-6038 EMAIL: [email protected] Mr. Joseph Barker, Assistant Division Chair Ms. Annette Orrico, Assistant Division Chair TEL: SC (708) 579-6581, NC (708) 579-6412 TEL: SC: (708) 579-6583, NC: (708) 579-6411 FAX: (708) 579-6038 FAX: (708) 579-6038 EMAIL: [email protected] EMAIL: [email protected] Mathematics Department Philosophy Th e mathematics curriculum has been developed to help students value mathematics, become confi dent in their abilities to do mathematics, become mathematical problem solvers, and to communicate and reason mathemati- cally. Students, as a result of the high school mathematics experiences, should be able to model problems with the appropriate operations and equations, apply a variety of approaches and techniques to solve problems, understand the underlying mathematical features of problems, see the applicability of mathematical ideas to common and complex problems, use logical reasoning to present an argument, and employ technology to explore mathematical ideas and solve problems. Students who successfully completed Algebra (Accel) in Grade 7th or 8th grade will receive one unit of high school credit on a pass/fail basis. Th e high school credit will be awarded aft er successful completion of one year of mathematics while enrolled in high school. Mathematics 102 Mathematics Department Standards The LTHS Mathematics Department has adopted the following eight principles in conjunction with both the Illinois State Standards and the Common Core State Standards.
[Show full text]
• Abstract Algebra Paul Garrett
- Abstract Algebra Paul Garrett ii I covered this material in a two-semester graduate course in abstract algebra in 2004-05, rethinking the material from scratch, ignoring traditional prejudices. I wrote proofs which are natural outcomes of the viewpoint. A viewpoint is good if taking it up means that there is less to remember. Robustness, as opposed to fragility, is a desirable feature of an argument. It is burdensome to be clever. Since it is non-trivial to arrive at a viewpoint that allows proofs to seem easy, such a viewpoint is revisionist. However, this is a good revisionism, as opposed to much worse, destructive revisionisms which are nevertheless popular, most notably the misguided impulse to logical perfection [sic]. Logical streamlining is not the same as optimizing for performance. The worked examples are meant to be model solutions for many of the standard traditional exercises. I no longer believe that everyone is obliged to redo everything themselves. Hopefully it is possible to learn from others’ eﬀorts. Paul Garrett June, 2007, Minneapolis Garrett: Abstract Algebra iii Introduction Abstract Algebra is not a conceptually well-deﬁned body of material, but a conventional name that refers roughly to one of the several lists of things that mathematicians need to know to be competent, eﬀective, and sensible. This material ﬁts a two-semester beginning graduate course in abstract algebra. It is a how-to manual, not a monument to traditional icons. Rather than an encyclopedic reference, it tells a story, with plot-lines and character development propelling it forward. The main novelty is that most of the standard exercises in abstract algebra are given here as worked examples.
[Show full text]