Rational Numbers To Linear Equations

This is the first of three volumes that, together, give an exposition of the mathematics of grades 9–12 that is simultaneously mathematically correct and grade-level appropriate. The volumes are consistent with CCSSM (Common Core State Standards for Mathematics) and aim at presenting the mathematics of K–12 as a totally transparent subject. The present volume begins with fractions, then rational numbers, then introductory geometry that can make sense of the slope of a line, then an explanation of the correct use of symbols that makes sense of “variables”, and finally a systematic treatment of linear equations that explains why the graph of a linear equation in two variables is a straight line and why the usual solution method for simultaneous linear equations “by substitutions” is correct. This book should be useful for current and future teachers of K–12 mathematics, as well as for some high school students and for education professionals.

"This book is awesome! What stood out to me was the deep understanding I was able to have about what fluency actually means. Too often the message has been fluency and accuracy, especially at the middle school level. By providing teachers with tools for building fluency with integers, expressions, and algebra, this book shifts that message to also focus on flexibility and strategy selection." Lindsey Henderson Secondary Mathematics Specialist, Utah State Board of Education Salt Lake City, UT Because fluency practice is not a worksheet. Fluency in mathematics is more than adeptly using basic facts or implementing algorithms. It is not about speed or recall. Real fluency is about choosing strategies that are efficient, flexible, lead to accurate solutions, and are appropriate for the given situation. Developing fluency is also a matter of equity and access for all learners. The landmark book Figuring Out Fluency in Mathematics Teaching and Learning offered educators the inspiration to develop a deeper understanding of procedural fluency, along with a plethora of pragmatic tools for shifting classrooms toward a fluency approach. Now, teachers have the chance to apply that inspiration through explicit instruction and practice every day with the classroom companion Figuring Out Fluency—Operations With Rational Numbers and Algebraic Equations. With this book, teachers can Dive deeper into the Significant Strategies for fluency explained in the anchor book as they apply to rational number operations Explore how these strategies can be applied for proportional reasoning, solving equations for unknowns, and solving systems of linear equations Access over 100 classroom-ready activities, including worked examples, routines, and games. Find activities to explicitly teach students how to use and choose strategies to operate on rational numbers and solve algebraic equations Download all of the needed support tools, game boards, and other resources from the companion website for immediate implementation Give each and every student the knowledge and power to become skilled and confident mathematical thinkers and doers.

Transforming the standards into learning outcomes just got a lot easier In this resource, you can see in an instant how teaching to your state standards should look and sound in the classroom. Under the premise that math is math, the authors provide a Cross-Referencing Index for states implementing their own specific mathematics standards, allowing you to see and understand which page number to turn to for standards-based teaching ideas. It’s all here, page by page: Get the inside scoop on which standards connect, what key vocabulary means, and time-saving tables showing where to focus instruction for each grade Write curriculum for: ratios and proportional relationships, the number system, expressions and equations, functions, geometry, and statistics & probability Use the What to Teach pages to deliver powerful standards-based lessons Learn effective techniques to create an environment where all students can experience math break-throughs Incorporate the Standards for Mathematical Practice to improve students’ ability to problem solve, construct viable arguments, use tools strategically, attend to precision, and more Cross-referenced index listing the standards in the following states, explaining what is unique to the standards of each state Your Mathematics Standards Companion is your one-stop guide for teaching, planning, assessing, collaborating, and designing powerful mathematics curriculum.

This book explores the rich history of community college math with a specific focus on gatekeeper math classes. Gatekeeper math classes include courses such as college algebra, introduction to statistics, and all developmental math classes. For community colleges, successful completion of these classes is imperative for student retention. This book presents a decade-by-decade analysis of the history of community college mathematics. The author employs a mix of conceptual, empirical, and quantitative research. The empirical research stems from interviews with 30 community college faculty members from seven community colleges. From the 1970s to the pandemic in the early 2020s, the book explores math curricula as well as trends, initiatives, teaching practices, and mandates that have impacted community college math. The positives and negatives of such trends, initiatives, and mandates are presented along with suggestions on how to apply such knowledge going forward. The author addresses the key questions: How can we build a future model for community college gatekeeper math classes that is both successful and sustainable? Additionally, how can we learn from the past and the present to build such a model? This book will be ideal for students in graduate programs focusing on community college leadership or developmental education leadership as well as all those hoping to improve success rates in community college mathematics programs.

This book is written as an introduction to the theory of error-free computation. In addition, we include several chapters that illustrate how error-free com putation can be applied in practice. The book is intended for seniors and first year graduate students in fields of study involving scientific computation using digital computers, and for researchers (in those same fields) who wish to obtain an introduction to the subject. We are motivated by the fact that there are large classes of ill-conditioned problems, and there are numerically unstable algorithms, and in either or both of these situations we cannot tolerate rounding errors during the numerical computations involved in obtaining solutions to the problems. Thus, it is important to study finite number systems for digital computers which have the property that computation can be performed free of rounding errors. In Chapter I we discuss single-modulus and multiple-modulus residue number systems and arithmetic in these systems, where the operands may be either integers or rational numbers. In Chapter II we discuss finite-segment p-adic number systems and their relationship to the p-adic numbers of Hensel [1908]. Each rational number in a certain finite set is assigned a unique Hensel code and arithmetic operations using Hensel codes as operands is mathe matically equivalent to those same arithmetic operations using the cor responding rational numbers as operands. Finite-segment p-adic arithmetic shares with residue arithmetic the property that it is free of rounding errors.