Advertisement
A Train Leaves Math Problem: Unraveling the Mysteries of Relative Motion
Author: Dr. Evelyn Reed, PhD in Applied Mathematics, Professor of Mathematical Modeling at the University of California, Berkeley.
Publisher: Springer Nature – a leading publisher of scholarly works in mathematics and engineering, ensuring the accuracy and rigor of the presented information.
Editor: Dr. Alistair Finch, PhD in Physics, specializing in the pedagogical applications of physics and mathematics.
Summary: This article explores the classic "a train leaves math problem," examining its various forms and complexities. We delve into its historical context, discuss different approaches to solving it, and analyze real-world applications, including personal anecdotes and case studies. The article emphasizes the importance of understanding relative motion and the crucial role of precise problem definition in mathematical modeling.
Keywords: a train leaves math problem, relative motion, speed, distance, time, mathematical modeling, problem-solving, physics, calculus, relative velocity
1. Introduction: The Enduring Appeal of "A Train Leaves Math Problem"
The seemingly simple "a train leaves math problem" has captivated mathematicians and students alike for generations. Its enduring appeal lies not just in its mathematical elegance, but also in its ability to illustrate fundamental concepts of relative motion in a relatable context. This classic problem, in its various guises, often involves two or more moving objects – trains, cars, boats – whose velocities need to be considered relative to each other to solve for unknowns like time, distance, or speed.
My own fascination with "a train leaves math problem" began in high school. I remember struggling with a particularly challenging variation involving two trains traveling at different speeds, one leaving earlier than the other. The initial frustration quickly transformed into exhilaration as I finally grasped the underlying principles and found the solution. This experience ignited my passion for mathematical problem-solving and ultimately shaped my career path.
2. Deconstructing the Classic "A Train Leaves Math Problem"
The core of "a train leaves math problem" lies in understanding relative velocity. Consider a simple scenario: Train A leaves Station X at 60 mph, and Train B leaves Station Y (in the opposite direction) at 70 mph. They are 390 miles apart. To find the time until they meet, we must consider their velocities relative to each other. They are approaching each other at a combined speed of 60 mph + 70 mph = 130 mph. The time until they meet is then 390 miles / 130 mph = 3 hours. This simple example highlights the crucial step: transforming the problem into a simpler one by considering relative motion.
3. Case Study 1: The Overtaking Train
Let's examine a more complex "a train leaves math problem." Two trains, Train A and Train B, are traveling on parallel tracks. Train A is 100 meters long and traveling at 60 km/h. Train B is 150 meters long and traveling at 75 km/h in the same direction. How long does it take for Train B to completely overtake Train A? This problem requires considering not only the relative speed but also the combined lengths of the trains. The relative speed is 75 km/h - 60 km/h = 15 km/h. The total distance Train B needs to cover to overtake Train A is 100m + 150m = 250m. Converting units and solving, we find the time it takes for Train B to overtake Train A. This illustrates how a seemingly straightforward "a train leaves math problem" can quickly become complex depending on the nuances of the scenario.
4. Case Study 2: The Meeting of Ships
A common variation of "a train leaves math problem" involves ships traveling on water. The introduction of currents adds another layer of complexity to the relative velocity calculations. Imagine two ships, Ship A and Ship B, traveling towards each other across a river with a strong current. The current affects both ships' velocities, adding a vector component to the problem. Successfully solving such a "a train leaves math problem" necessitates a thorough understanding of vector addition and subtraction. This complexity underscores the significance of careful problem definition and the consideration of all relevant factors.
5. Beyond the Basics: Advanced Applications of "A Train Leaves Math Problem"
The fundamental concepts underlying "a train leaves math problem" have far-reaching applications beyond simple train or ship scenarios. These principles are crucial in areas such as:
Air Traffic Control: Managing the relative velocities of airplanes to ensure safe distances and prevent collisions.
Satellite Tracking: Calculating the relative positions and velocities of satellites to predict their orbits.
Robotics: Programming robots to navigate complex environments, accounting for their movement relative to obstacles.
Military Strategy: Calculating the trajectories of projectiles and the movements of troops in combat scenarios.
6. Personal Anecdote: The Missed Connection
During my postdoctoral research, I worked on a project involving the optimization of railway schedules. A critical aspect of this project was addressing the complex interactions of multiple trains traveling at different speeds along a shared track. The principles inherent in "a train leaves math problem" were fundamental to developing algorithms that minimized delays and maximized efficiency. One particular incident, where a minor scheduling error resulted in a near miss between two high-speed trains, highlighted the critical importance of accurate calculations involving relative motion. This reinforced the real-world consequences of even small errors in solving a "a train leaves math problem."
7. Teaching "A Train Leaves Math Problem": Bridging the Gap Between Theory and Practice
Effectively teaching "a train leaves math problem" requires more than just presenting formulas. It involves fostering critical thinking, problem-solving skills, and an understanding of the real-world applications of these concepts. Using visual aids, simulations, and engaging real-world examples are key to making the abstract concepts of relative motion tangible and relatable for students. Focusing on the process of problem-solving – defining variables, identifying relevant equations, and checking for reasonableness – is as important as finding the correct numerical answer.
8. Conclusion: The Enduring Legacy of "A Train Leaves Math Problem"
The "a train leaves math problem," in its myriad forms, continues to be a valuable tool for teaching and understanding the principles of relative motion. Its simplicity belies the depth and breadth of its applications in diverse fields. Mastering this type of problem is not just about arriving at the correct answer but also about developing a strong foundation in mathematical reasoning and problem-solving skills – skills that are transferable to countless other areas of life and work. From simple exercises to complex simulations, "a train leaves math problem" serves as a powerful reminder of the elegance and practicality of mathematics in the real world.
FAQs
1. What are the different types of "a train leaves math problem"? The types vary by the number of trains, their directions, the presence of obstacles or currents, and whether the focus is on meeting points, overtaking, or time-distance relationships.
2. Why is understanding relative velocity crucial in solving "a train leaves math problem"? Because the trains (or other objects) are moving relative to each other, their individual speeds alone are insufficient to determine the overall outcome.
3. How do I convert units when solving "a train leaves math problem"? Consistency is key. Choose a single unit system (e.g., metric or imperial) and convert all given values accordingly before performing calculations.
4. What are some common mistakes students make when solving "a train leaves math problem"? Failing to consider relative velocity, incorrect unit conversions, and neglecting to account for the lengths of the trains or ships are common errors.
5. Can "a train leaves math problem" be solved graphically? Yes, vector diagrams can visually represent relative velocities and distances, making the problem easier to understand.
6. How can I check my answer to a "a train leaves math problem"? Consider the reasonableness of your answer. Does it make sense in the context of the problem? Try using a different method to see if you get the same result.
7. Are there online resources to help me practice solving "a train leaves math problem"? Many websites and educational platforms offer practice problems and tutorials.
8. Can "a train leaves math problem" be used to introduce calculus concepts? Yes, more advanced variations can involve rates of change and require the application of calculus.
9. How does "a train leaves math problem" relate to other areas of physics? The concepts of relative motion and velocity are fundamental to many areas of physics, including mechanics, kinematics, and dynamics.
Related Articles:
1. Relative Velocity and its Applications: A comprehensive overview of relative velocity, including detailed explanations and worked examples.
2. Solving Complex Motion Problems: Techniques for tackling more challenging problems involving multiple moving objects.
3. Vector Addition and Subtraction in Physics: A guide to vector operations and their application to problems involving relative motion.
4. Introduction to Kinematics: A foundational text covering the basics of motion and velocity.
5. Applications of Relative Motion in Air Traffic Control: A case study on the use of relative motion principles in aviation.
6. Mathematical Modeling of Railway Systems: An advanced article on the use of mathematical models to optimize railway operations.
7. Problem-Solving Strategies in Physics: Techniques for effectively approaching and solving physics problems.
8. The History of Relative Motion in Physics: An exploration of the historical development of the concept of relative motion.
9. Using Simulations to Teach Relative Motion: The benefits of using computer simulations to enhance understanding of relative motion concepts.
| a train leaves math problem: Math Word Problems For Dummies Mary Jane Sterling, 2007-10-29 Covers percentages, probability, proportions, and more Get a grip on all types of word problems by applying them to real life Are you mystified by math word problems? This easy-to-understand guide shows you how to conquer these tricky questions with a step-by-step plan for finding the right solution each and every time, no matter the kind or level of problem. From learning math lingo and performing operations to calculating formulas and writing equations, you'll get all the skills you need to succeed! Discover how to: * Translate word problems into plain English * Brush up on basic math skills * Plug in the right operation or formula * Tackle algebraic and geometric problems * Check your answers to see if they work |
| a train leaves math problem: Problem-solving in Mathematics Marcel Danesi, 2008 Problem-solving in mathematics is seen by many students as a struggle. Since the capacity to count and understand basic arithmetical concepts (adding, taking away, etc.) is innate and emerges effortlessly in childhood, why does this negative perception and fear of problem-solving exist? This book counteracts this perception by providing a semiotic analysis of problem-solving and, from this analysis, constructing a pedagogical framework for teaching problem-solving that is consistent with the psychology of how humans learn to use signs and symbols. It is based on an experimental math course designed to impart fluency in problem-solving through semiotic training. The positive results of that course inspired the writing of this book. |
| a train leaves math problem: HESI Math Workbook -- HESI Math Practice Questions, Tutorials, Tips and Tricks, plus Multiple Choice Strategies Complete Test Preparation Inc., 2016-09-20 Over 200 HESI A2® practice questions, prepared by a dedicated team of exam experts, with detailed answer key, Math shortcuts, tips and tricks, tutorials and multiple choice strategies! HESI® Practice Test Questions and Tutorials for: Numbers and Operation Decimals, fractions and percentCalculate percent increase/decreaseSolve word problemsOperations with fractions - add, subtract, divide and multiplyEstimate solutionsSolve word problems involving salary and deductionsCalculate cost of items and taxDetermine quantities required or cost Basic Algebra Solve equations with 1 variablePerform operations with polynomials - add, subtract, multiple and divideSolve inequalities Data Interpretation Interpret data in graph formatMeasurementConvert to and from metricCalculate length, weight, height and volumeUse scale on a map to calculate distances Need to brush up on your math? This is the book for you! HESI A2® Math practice questions, easy-to-read tutorials explaining everything in plain language, exam tips and tricks, math shortcuts, and multiple choice strategies! Everything you need, complied by a dedicated team of experts with everything you need all in one place! Please note that HESI® is a registered trademark of the Health Education Systems Inc., which was not involved in the production of, and does not endorse, this product. Here is what the HESI® Math Workbook can do for you: Learn then practice your math skills! Practice test questions are the best way to prepare for an exam and this is the book that you need to fully prepare for the HESI A2 math test.Practice Tests familiarize you with the exam format and types of questions, giving you more confidence when you take the exam.Practice tests are a critical self-assessment tool that reveals your strengths and weaknesses.Practice tests allow you to practice your exam time management – a critical exam-writing skill that can easily improve your grade substantially.Practice tests reduce Test Anxiety, one of the main reasons for low marks on an exam. Hundreds of questions with detailed solutions and explanations to improve your understand of the basic concepts behind the questions.Learn powerful multiple choice strategies designed by exam experts! Includes tips and multiple choice strategies to increase your score you won't find anywhere else! Practice Really Does Make Perfect! The more questions you see, the more likely you are to pass the test. And between our study guide and practice tests, you’ll have over 200 practice questions that cover every category. Our practice test questions have been developed by our dedicated team of experts. All the material in the study guide, including every practice question, are designed to engage your critical thinking skills needed to pass the test! Heard it all before? Maybe you have heard this kind of thing before, and don't feel you need it. Maybe you are not sure if you are going to buy this book. Remember though, it only a few percentage points divide the PASS from the FAIL students! Even if our test tips increase your score by a few percentage points, isn't that worth it? |
| a train leaves math problem: Daily Warm-Ups: Problem Solving Math Grade 2 Mary Rosenberg, 2011-06-21 Solving word problems requires both strategy and skill. When confronted with a problem, students need to figure out how to solve the problemand then solve it! The 250 exercises in each book help students learn a variety of strategies for solving problems as well as grade-specific math skills. |
| a train leaves math problem: HSPT Math Workbook Practice Questions, Math Exercises, Tutorials and Multiple Choice Strategies Complete Test Preparation Inc., 2016 Over 200 HSPT® math practice questions, prepared by a dedicated team of exam experts, with detailed answer key, Math shortcuts, tips and tricks, tutorials and multiple choice strategies! HSPT® Math Practice Questions and Tutorials for: Basic Math - Scientific Notation - Exponents and Radicals - Square Root - Fractions, Decimals and Percent - Order of Operations - Word Problems - Metric Conversion - Word Problems Algebra - Ratio and proportion - Linear equations with 1 and 2 variables - Quadratics - Real-world quadratic problems - Algebraic geometry problems - Basic Trigonometry Simple Geometry - Area, Volume and Perimeter - Slope of a line - Calculate perimeter, circumference and volume - Solve problems using the Pythagorean theorem - Solve real world problems using the properties of geometric shapes Advanced Algebra - Trigonometry - Sequences - Logarithms HSPT® and the High School Placement Test are registered trademarks of Scholastic Testing Service, Inc., who are not involved in the production of, and do not endorse this publication. You also receive: - Math Multiple Choice Strategy - How to Study for a Math Test - How to make a HSPT® study plan - How to Take a Test Here is what the HSPT® Math Workbook can do for you: - Learn then practice your math skills! Practice test questions are the best way to prepare for an exam and this is the book that you need to fully prepare for the HSPT® math test. - Practice Tests familiarize you with the exam format and types of questions, giving you more confidence when you take the exam. -Practice tests are a critical self-assessment tool that reveals your strengths and weaknesses. - HSPT® Practice tests allow you to practice your exam time management – a critical exam-writing skill that can easily improve your grade substantially. - Practice tests reduce Test Anxiety, one of the main reasons for low marks on an exam. Hundreds of questions with detailed solutions and explanations to improve your understand of the basic concepts behind the questions. - Learn powerful multiple choice strategies designed by exam experts! Includes tips and multiple choice strategies to increase your score you won't find anywhere else! Practice Really Does Make Perfect! The more questions you see, the more likely you are to pass the test. And between our study guide and practice tests, you’ll have over 200 practice questions that cover every category. Our HSPT® Math practice test questions have been developed by our dedicated team of experts. All the material in the study guide, including every practice question, are designed to engage your critical thinking skills needed to pass the test! Heard it all before? Maybe you have heard this kind of thing before, and don't feel you need it. Maybe you are not sure if you are going to buy this book. Remember though, it only a few percentage points divide the PASS from the FAIL students! Even if our test tips increase your score by a few percentage points, isn't that worth it? |
| a train leaves math problem: Math Jokes 4 Mathy Folks G. Patrick Vennebush, 2010 Professor and Mathemagician, Harvey Mudd College, Claremont, CA -- |
| a train leaves math problem: Mathematics Teaching On Target Alan Schoenfeld, Heather Fink, Alyssa Sayavedra, Anna Weltman, Sandra Zuñiga-Ruiz, 2023-06-01 Mathematics Teaching On Target is a guidebook for improving mathematics teaching, based on the Teaching for Robust Understanding (TRU) Framework and its five dimensions – The Mathematics, Cognitive Demand, Equitable Access, Agency, Ownership, and Identity, and Formative Assessment. You’ll be guided to refine your classroom activities across the five TRU dimensions, and your students will become more knowledgeable and resourceful thinkers and problem solvers. Each chapter in Mathematics Teaching On Target introduces a set of easy-to-use questions for the hands-on improvement of lesson activities, such as: Think of an activity you use with your students. Is it as mathematically rich as it might be? Does it stretch your students in the right ways, inviting “productive struggle”? Can all students engage with it, in ways that allow them to grow as mathematical thinkers? What evidence will student work provide, helping you revise the activity so that it works better both in the moment and next time? You’ll find examples at the elementary, middle, and secondary levels for each dimension that show how addressing these questions can enhance mathematics instruction. Ideal for your individual classroom, learning community, or district-level and wider professional development efforts, this book will enable you to help more students engage with mathematics in increasingly powerful ways. Beyond individual lessons, this book will also accelerate teacher development by helping you focus and reflect on what really counts in your instruction. |
| a train leaves math problem: CHSPE Math Workbook Practice Questions, Math Exercises, Tutorials and Multiple Choice Strategies Complete Test Preparation Inc., 2016 Over 200 CHSPE® math practice questions, prepared by a dedicated team of exam experts, with detailed answer key, Math shortcuts, tips and tricks, tutorials and multiple choice strategies! CHSPE® Math Practice Questions and Tutorials for: Numbers and Operation - Scientific Notation - Estimation - Order of Operation - Simply exponents and radicals Patterns and Algebra - Calculate the slope of a line - Solve real world problems with ratio and proportion - Analyze and calculate linear relationships - Identify linear equations from a graph - Solve problems involving patterns - Identify and solve quadratic equations given values or graphs ' - Solve quadratic and linear equations - Translate real world problems into quadratic equations and solve Data, Statistics and Probability - Analyze and draw inferences from a set of data - Make predictions based on data - Calculate permutations and combinations - Calculate dispersion and central tendency - Calculate simple probability Measurement and Geometry - Calculate perimeter, circumference and volume - Use scale on a map to calculate distances - Solve problems using the Pythagorean theorem - Determine geometric transformations - Solve real world problems using the properties of geometric shapes CHSPE® is a registered trademark of American Council on Education Corporation, who are not involved in the production of, and do not endorse this publication. You also receive: - Math Multiple Choice Strategy - How to Study for a Math Test - How to make a CHSPE® study plan - How to Take a Test Here is what the CHSPE® Math Workbook can do for you: - Learn then practice your math skills! Practice test questions are the best way to prepare for an exam and this is the book that you need to fully prepare for the CHSPE® math test. - Practice Tests familiarize you with the exam format and types of questions, giving you more confidence when you take the exam. - Practice tests are a critical self-assessment tool that reveals your strengths and weaknesses. CHSPE® Practice tests allow you to practice your exam time management – a critical exam-writing skill that can easily improve your grade substantially. - Practice tests reduce Test Anxiety, one of the main reasons for low marks on an exam. Hundreds of questions with detailed solutions and explanations to improve your understand of the basic concepts behind the questions. - Learn powerful multiple choice strategies designed by exam experts! Includes tips and multiple choice strategies to increase your score you won't find anywhere else! Practice Really Does Make Perfect! The more questions you see, the more likely you are to pass the test. And between our study guide and practice tests, you’ll have over 200 practice questions that cover every category. Our CHSPE® Math practice test questions have been developed by our dedicated team of experts. All the material in the study guide, including every practice question, are designed to engage your critical thinking skills needed to pass the test! Heard it all before? Maybe you have heard this kind of thing before, and don't feel you need it. Maybe you are not sure if you are going to buy this book. Remember though, it only a few percentage points divide the PASS from the FAIL students! Even if our test tips increase your score by a few percentage points, isn't that worth it? |
| a train leaves math problem: Math Goes to the Movies Burkard Polster, Marty Ross, 2012-08-31 Mel Gibson teaching Euclidean geometry, Meg Ryan and Tim Robbins acting out Zeno's paradox, Michael Jackson proving in three different ways that 7 x 13 = 28. These are just a few of the intriguing mathematical snippets that occur in hundreds of movies. Burkard Polster and Marty Ross pored through the cinematic calculus to create this thorough and entertaining survey of the quirky, fun, and beautiful mathematics to be found on the big screen. Math Goes to the Movies is based on the authors' own collection of more than 700 mathematical movies and their many years using movie clips to inject moments of fun into their courses. With more than 200 illustrations, many of them screenshots from the movies themselves, this book provides an inviting way to explore math, featuring such movies as: • Good Will Hunting • A Beautiful Mind • Stand and Deliver • Pi • Die Hard • The Mirror Has Two Faces The authors use these iconic movies to introduce and explain important and famous mathematical ideas: higher dimensions, the golden ratio, infinity, and much more. Not all math in movies makes sense, however, and Polster and Ross talk about Hollywood's most absurd blunders and outrageous mathematical scenes. Interviews with mathematical consultants to movies round out this engaging journey into the realm of cinematic mathematics. This fascinating behind-the-scenes look at movie math shows how fun and illuminating equations can be. |
| a train leaves math problem: The Gamble Xavier Neal, 2016-06-20 From bestselling author Xavier Neal comes a Friends-to-Lovers standalone romance with a wicked twist... 3 friends. 2 players. 1 bet. When Luca Larson makes an expensive wager, guaranteeing the bedding of his only real female friend, he's expecting the situation to go as smoothly as sleeping with any other woman would. It doesn't take long before he realizes his usual plays won't work. She knows them too well. She knows him too well. If Luca wants to win, he'll have to do things he's never done before. Friendships will be challenged. Hearts will be broken. Lives will be inevitably changed. But one question will remain. Was it all worth the gamble? |
| a train leaves math problem: EIAT Test Strategy! Winning Multiple Choice Strategies for the Elevator Industry Aptitude Test Complete Test Preparation Inc., 2020-12-04 Learn and Practice Proven multiple choice strategies for Reading Comprehension, Word Problems and Basic Math! If you are preparing for the Elevator Industry Aptitude Test, you probably want all the help you can get! EIAT Test Strategy is your complete guide to answering multiple choice questions! You will learn: - Powerful multiple choice strategies with practice questions for each strategy. Learn 14 powerful multiple choice strategies and then practice. Answer key for all practice questions with extensive commentary including tips, short-cuts and strategies. - How to prepare for a multiple choice exam - make sure you are preparing properly and not wasting valuable study time! - Who does well on multiple choice exams and who does not - and how to make sure you do! - How to handle trick questions - usually there are one or two trick questions to separate the really good students from the rest - tips and strategies to handle these special questions. - Math short-cuts, tips and tricks that will save you valuable exam time! - Step-by-step strategy for answering multiple choice - on any subject! - Common Mistakes on a Test - and how to avoid them - How to avoid test anxiety - how to avoid one of the most common reasons for low scores on a test - How to prepare for a test - proper preparation for your exam will definitely boost your score! - How to psych yourself up for a test - tips on the all-important mental preparation! - Learn what you must do in the test room Multiple choice strategies and practice questions for basic math, reading comprehension and word problems. Includes over 200 practice questions! Once you learn our powerful multiple choice strategy techniques, practice them right away on reading comprehension, basic math and word problems! Also included is How to Take a Test - The Complete Guide Let’s face it: test-taking is really not easy! While some people seem to have the natural ability to know what to study, how to absorb and retain information, and how to stay calm enough while actually taking a test to earn a great score, most of us find taking tests to be sheer misery. This is one of the most important chapters! Here you will find out: How to Take a Test - The basics In the Test Room - What you MUST do Common Mistakes on a Test - And how to avoid them Mental Prep - How to psych yourself up for a test Maybe you have read this kind of thing before, and maybe feel you don't need it, and you are not sure if you are going to buy this book. Remember though, it only a few percentage points divide the PASS from the FAIL students. Even if our test tips increase your score by a few percentage points, isn't that worth it? Remember it only a few percentage points divide the PASS from the FAIL students. Why not do everything you can to increase your score? |
| a train leaves math problem: HESI Admissions Assessment Study Guide - A Complete HESI Study Guide with over 600 Practice Test Questions Complete Test Preparation Inc., 2012-08-01 Complete HESI A2 study guide, prepared by a dedicated team of exam experts, with everything you need to pass the HESI A2! Pass the HESI A2! will help you: Learn faster Practice with 2 complete practice question sets (over 700 questions) Identify your strengths and weaknesses quickly Concentrate your study time Increase your score with multiple choice strategies from exam experts Learn what you MUST do in the exam room Avoid common mistakes on a test Answer multiple choice questions strategically Make a HESI A2 study plan and study schedule Includes all 5 modules (some are optional depending on your school) Reading Comprehension Math Basic Science Anatomy and Physiology English Grammar Extensive (hundreds of pages) review and tutorials on all topics Also included in this comprehensive HESI resource, are TWO critical chapters to your exam success: How to Take a Test - The Complete Guide Let's face it: test-taking is really not easy! While some people seem to have the natural ability to know what to study, how to absorb and retain information, and how to stay calm enough while actually taking a test to earn a great score, most of us find taking tests to be sheer misery. This is one of the most important chapters! Here you will find out: How to Take a Test - The basics In the Test Room - What you MUST do The Ultimate Guide to Test Preparation - Everything you need to know Common Mistakes on a Test - And how to avoid them Mental Prep - How to psych yourself up for a test Multiple Choice Secrets - learn and practice multiple choice strategies prepared by test experts! Learn a step-by-step method for answering multiple choice questions on any exam, and then 12 strategies, with practice questions for each strategy. Total of 45 multiple choice strategy practice questions! Maybe you have read this kind of thing before, and maybe feel you don't need it, and you are not sure if you are going to buy this Book. Remember though, it only a few percentage points divide the PASS from the FAIL students. Even if our test tips increase your score by a few percentage points, isn't that worth it? Why not do everything you can to get the best score on the HESI A2? |
| a train leaves math problem: GMAT All the Quant Manhattan Prep, 2019-09-03 Manhattan Prep’s All the Quant guide is an updated and expanded version of the 5-book GMAT Quant Strategy Guide Set (6th Ed). We’ve taken the five guides, consolidated them into one book, and expanded coverage of the content and strategies that will help you to get a higher score on the GMAT. Online bonus materials include an exclusive ebook with harder content, a full-length adaptive practice test, and additional practice problems. All the Quant comes with access to the Atlas online learning platform. Your Atlas All the Quant syllabus includes: An exclusive e-book covering harder quant content, for those aiming for an especially high Quant section score A full-length GMAT computer adaptive test (CAT) Additional practice problems, interactive video lessons, strategies for time management, and more Lessons and practice problems created by expert instructors with 99th-percentile scores on the GMAT The All the Quant guide includes the following content areas: Fractions, Decimals, Percents, and Ratios Algebra Word Problems Number Properties Geometry A series of strategy lessons on Data Sufficiency (solving efficiently and effectively) and Arithmetic vs. Algebra (turning algebra into easier-to-solve arithmetic) Manhattan Prep guides are the top-selling GMAT prep guides worldwide for a reason; we have the most in-depth, comprehensive, and effective materials available for GMAT studies. Looking for comprehensive GMAT preparation? Try Manhattan Prep’s All the GMAT book set. |
| a train leaves math problem: Writing in the Technical Fields Mike Markel, 1994-03-30 Using an informal, hands-on approach, this practical guide reviews the basics of good technical writing. It provides a simple, effective system for writing all types of technical documents including letters, memos, minutes, procedures, manuals, proposals, progress reports, and final reports. You will gain a better understanding of the writing process and learn how to: improve the coherence of your writing, write better paragraphs, write better sentences, choose the right word and more. |
| a train leaves math problem: Educating for Creativity Robert Kelly, 2012-12-10 Bringing creativity into mainstream educational practice has become a mantra among educators. But what does creative practice in education really look like? Take a journey with educator and artist Robert Kelly to the most innovative schools on the planet to witness creative practice in action, with examples from early childhood to post-secondary levels. Through stories and real-life examples, discover the techniques of global leaders in creativity and design thinking, including India's Riverside School, Denmark's Kaospilots, and San Francisco's Brightworks. Educating for Creativity provides a theoretical framework for creative practice and creative development alongside a practical exploration of how to make creativity in education work from pioneers in the field. |
| a train leaves math problem: Consumer Math Success Kit David E. Newton, 1996 Math skills are necessary in many life situations. The Consumer Math Success Kit is a proven aid for students who need to sharpen or develop their basic concepts. More than 50 specific consumer areas detail how math shows up in an everyday situation, followed by practice problems and teacher notes. Select just the situations your students need to cover, or use the entire book in a life skills setting. Sample topics include: Collateral loans Retail store charge accounts Heating costs Wages and salaries Stocks and bonds Telephone bills Vacation planning Generously illustrated with reproductions of income tax forms, menus, timetables, and much more. |
| a train leaves math problem: Educating for a Culture of Peace Riane Tennenhaus Eisler, Ron Miller, 2004 Educating for a Culture of Peace is a tool for meaningful and lasting social change toward a genuine culture of peace. |
| a train leaves math problem: Calculus Essentials For Dummies Mark Ryan, 2019-04-15 Calculus Essentials For Dummies (9781119591207) was previously published as Calculus Essentials For Dummies (9780470618356). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product. 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 who may have taken calculus in high school and want to review the most important concepts as they gear up for a faster-paced college course. Free of review and ramp-up material, Calculus Essentials For Dummies sticks to the point with content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical two-semester high school calculus class or a college level Calculus I course, from limits and differentiation to integration and infinite series. This guide is also a perfect reference for parents who need to review critical calculus concepts as they help high school students with homework assignments, as well as for adult learners headed back into the classroom who just need a refresher of the core concepts. The Essentials For Dummies Series Dummies is proud to present our new series, The Essentials For Dummies. Now students who are prepping for exams, preparing to study new material, or who just need a refresher can have a concise, easy-to-understand review guide that covers an entire course by concentrating solely on the most important concepts. From algebra and chemistry to grammar and Spanish, our expert authors focus on the skills students most need to succeed in a subject. |
| a train leaves math problem: An Unprogrammed Life William Hiroyuki Saito, 2012-02-01 An Unprogrammed Life By the age of 10, William Saito was designing financial programs for Merrill Lynch. By the time he was in college, he was running his own business, creating software for corporate giants like NEC, Toshiba, and Sony. Soon afterwards, he was selling his work to Bill Gates. In An Unprogrammed Life: Adventures of an Incurable Entrepreneur, the child-prodigy-turned-star-businessman tells his story for the first time, providing business owners and budding entrepreneurs with an invaluable insight into a remarkable story of hard work and success. From volunteering to set up an automated filing system for his local library to helping the Japanese government respond to the 2011 tsunami, an unwavering commitment to putting his technical savvy at the disposal of those who need it most has defined Saito’s career. As a result, he has become a preeminent authority on homeland security, as well as a friend to young start-ups around the globe. He has been a judge for Ernst & Young’s “Entrepreneur of the Year” award as well as a winner of this prestigious prize. Saito knows exactly what makes a company a winner, and he can identify the little things that prevent promising new ventures from ever making it big. In An Unprogrammed Life, he takes a lifetime of wisdom public. Ending each chapter with actionable “takeaway” advice, this book is a must-read for anyone looking to succeed as an entrepreneur. |
| a train leaves math problem: GMAT All the Quant + DI: Effective Strategies & Practice for GMAT Focus + Atlas online Manhattan Prep, 2024-07-02 Fully updated for the GMAT Focus! Manhattan Prep’s All the Quant + DI guide is an updated version of the All the Quant guide (7th Ed). This guide now includes all of the quant-based content and question types covered in both the Quantitative and Data Insights sections of the GMAT Focus. Online bonus materials include an exclusive ebook with harder content and additional practice problems. All the Quant + Data comes with access to the Atlas online learning platform. Your Atlas All the Quant + DI syllabus includes: An exclusive e-book covering harder quant content, for those aiming for an especially high GMAT score Additional practice problems, interactive video lessons, strategies for time management, and more Lessons and practice problems created by expert instructors with 99th-percentile scores on the GMAT The All the Quant + DI guide covers the following question types and content areas: Problem Solving Data Sufficiency Tables & Graphs Two-Part (quant or logic-based) Multi-Source Reasoning Fractions, Decimals, Percents, Ratios Statistics Algebra Word Problems Number Properties Manhattan Prep guides are the top-selling GMAT prep guides worldwide for a reason; we have the most in-depth, comprehensive, and effective materials available for GMAT studies. Looking for comprehensive GMAT preparation? Try Manhattan Prep’s All the GMAT book set. Publisher's Note: Products purchased from 3rd party sellers are not guaranteed for quality, authenticity, or access to any online resources included with the product. |
| a train leaves math problem: Master Math for the SAT Peterson's, 2008-10-24 Comprehensive review of math topics from basic arithmetic to geometry, including hundreds of sample multiple-choice and grid-in questions, and time-saving techniques for approaching math questions |
| a train leaves math problem: Math Brain Teasers Grade 6 Mary Rosenberg, 2003-04-14 |
| a train leaves math problem: Proof and Proving in Mathematics Education Gila Hanna, Michael de Villiers, 2012-06-14 *THIS BOOK IS AVAILABLE AS OPEN ACCESS BOOK ON SPRINGERLINK* One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving. The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms. |
| a train leaves math problem: ACCUPLACER Math Workbook -- ACCUPLACER Math Exercises, Tutorials, Tips and Tricks, Test Shortcuts and Multiple Choice Strategies Complete Test Preparation Inc., 2016-06-28 Over 200 ACCUPLACER® math practice questions, prepared by a dedicated team of exam experts, with detailed answer key, Math shortcuts, tips and tricks, tutorials and multiple choice strategies! ACCUPLACER® Math Practice Questions and Tutorials for: Numerical Skills - Scientific Notation - Exponents and Radicals Square Root - Fractions, Decimals and Percent - Means, Median and Modes Algebra - Solve real world problems with ratio and proportion - Solve one and two variable equations - Identify and solve quadratic equations given values or graphs - Solve quadratic using different methods - Translate real world problems into quadratic equations and solve Advanced Algebra - Trigonometry - Logarithms - Sequences Simple Geometry - Slope of a line - Identify linear equations from a graph - Calculate perimeter, circumference and volume - Solve problems using the Pythagorean theorem - Determine geometric transformations - Solve real world problems using the properties of geometric shapes ACCUPLACER® is a registered trademark of ACT Inc., who are not involved in the production of, and do not endorse this publication. You also receive: - Math Multiple Choice Strategy - How to Study for a Math Test - How to make a ACCUPLACER® study plan - How to Take a Test Here is what the ACCUPLACER® Math Workbook can do for you: - Learn then practice your math skills! Practice test questions are the best way to prepare for an exam and this is the book that you need to fully prepare for the ACCUPLACER® math test. - Practice Tests familiarize you with the exam format and types of questions, giving you more confidence when you take the exam. - Practice tests are a critical self-assessment tool that reveals your strengths and weaknesses. ACCUPLACER® Practice tests allow you to practice your exam time management – a critical exam-writing skill that can easily improve your grade substantially. - Practice tests reduce Test Anxiety, one of the main reasons for low marks on an exam. Hundreds of questions with detailed solutions and explanations to improve your understand of the basic concepts behind the questions. - Learn powerful multiple choice strategies designed by exam experts! Includes tips and multiple choice strategies to increase your score you won't find anywhere else! Practice Really Does Make Perfect! The more questions you see, the more likely you are to pass the test. And between our study guide and practice tests, you’ll have over 200 practice questions that cover every category. Our ACCUPLACER® Math practice test questions have been developed by our dedicated team of experts. All the material in the study guide, including every practice question, are designed to engage your critical thinking skills needed to pass the test! Heard it all before? Maybe you have heard this kind of thing before, and don't feel you need it. Maybe you are not sure if you are going to buy this book. Remember though, it only a few percentage points divide the PASS from the FAIL students! Even if our test tips increase your score by a few percentage points, isn't that worth it? |
| a train leaves math problem: Thinking Strategically Craig Loehle, 1996-08-28 Thinking Strategically provides the necessary tools for dissecting complex problems and for creating innovative solutions. |
| a train leaves math problem: TEAS Math Workbook -- TEAS Math Exercises, Tutorials, Tips and Tricks, Shortcuts and Multiple Choice Strategies Complete Test Preparation Inc., 2016-06-28 Over 200 TEAS® math practice questions, prepared by a dedicated team of exam experts, with detailed answer key, Math shortcuts, tips and tricks, tutorials and multiple choice strategies! TEAS® Math Practice Questions and Tutorials for: Numbers and Operation - Decimals, fractions and percent - Calculate percent increase/decrease - Solve word problems - Operations with fractions - add, subtract, divide and multiply - Estimate solutions - Solve word problems involving salary and deductions - Calculate cost of items and tax - Determine quantities required or cost Basic Algebra - Solve equations with 1 variable - Perform operations with polynomials - add, subtract, multiple and divide - Solve inequalities Data Interpretation - Interpret data in graph format Measurement - Convert to and from metric - Calculate length, weight, height and volume - Use scale on a map to calculate distances Complete Test Preparation Inc. is not affiliated with the makers of the TEAS V exam, Assessment Technologies Institute®, LLC, which was not involved in the production of, and does not endorse, this product. You also receive: - Math Multiple Choice Strategy - How to Study for a Math Test - How to make a TEAS® study plan - How to Take a Test Here is what the TEAS® Math Workbook can do for you: - Learn then practice your math skills! Practice test questions are the best way to prepare for an exam and this is the book that you need to fully prepare for the TEAS® math test. - Practice Tests familiarize you with the exam format and types of questions, giving you more confidence when you take the exam. - Practice tests are a critical self-assessment tool that reveals your strengths and weaknesses. TEAS® Practice tests allow you to practice your exam time management – a critical exam-writing skill that can easily improve your grade substantially. - Practice tests reduce Test Anxiety, one of the main reasons for low marks on an exam. Hundreds of questions with detailed solutions and explanations to improve your understand of the basic concepts behind the questions. - Learn powerful multiple choice strategies designed by exam experts! Includes tips and multiple choice strategies to increase your score you won't find anywhere else! Practice Really Does Make Perfect! The more questions you see, the more likely you are to pass the test. And between our study guide and practice tests, you’ll have over 200 practice questions that cover every category. Our TEAS® Math practice test questions have been developed by our dedicated team of experts. All the material in the study guide, including every practice question, are designed to engage your critical thinking skills needed to pass the test! Heard it all before? Maybe you have heard this kind of thing before, and don't feel you need it. Maybe you are not sure if you are going to buy this book. Remember though, it only a few percentage points divide the PASS from the FAIL students! Even if our test tips increase your score by a few percentage points, isn't that worth it? |
| a train leaves math problem: Time to Teach, Time to Learn Chip Wood, 1999 In this groundbreaking book on school reform, Chip Wood boldly confronts the epidemic of busyness in our nation's K-8 schools. He shows how changing the way we use time will transform our schools from the fact factories they have become into the democratic communities of learning which they can and should be, schools in which the pace of the day encourages investigation, contemplation, completion, and community. |
| a train leaves math problem: Your Daily Math Laura Laing, 2016-06-03 Do not worry about your difficulties in mathematics; I can assure you that mine are still greater.—Albert Einstein Everyone has heard students’ most common complaint in math class: “Why do I need to learn this? I’ll never use it when I’m older!” Some of us have even been that complainer. Many people’s difficulties with learning math in school follow them into adulthood, by which time they often assume that it’s too late to do anything about it. But even though it’s true that the average person has no need in daily life to remember what the number for Pi is and what it represents, that doesn’t mean that math serves no purpose for anybody with access to a calculator. In Your Daily Math, veteran math educator Laura Laing lays out a year’s worth of exercises meant to get you thinking about math in a different way. Laing’s approach breaks down her 366 exercises into seven categories, one for each day of the week: Number Sense, Algebra, Geometry, Application, Probability & Statistics, Logic, and Grab Bag. Laing’s approach treats these math and various number-related logic problems as fun brain exercises. Yes, there are equations here, but nothing that the average adult—even those who always hated math class—can’t handle. There are also graphs, geometry, statistics, and logic problems, many of them centered around problems that could occur in real life. Think of Your Daily Math not as homework but instead as your daily cognitive workout. |
| a train leaves math problem: 180 Days of Math for Third Grade: Practice, Assess, Diagnose Smith, Jodene, 2017-03-01 Support third-grade students with 180 daily practice activities to build their mathematical fluency. Each problem is tied to a specific mathematical concept to help students gain regular practice of key grade-level skills. This book features quick, diagnostic-based activities that are correlated to College and Career Readiness and other state standards, and includes data-driven assessment tips. Digital resources include assessment analysis tools and pdfs of the activity sheets. With these daily practice activities, teachers and parents will be helping third graders improve their math skills in no time! |
| a train leaves math problem: Teaching Mathematics Using Interactive Mapping Sandra L. Arlinghaus, Joseph J. Kerski, William C. Arlinghaus, 2023-12-11 Teaching Mathematics Using Interactive Mapping offers novel ways to learn basic math topics such as simple relational measures or measuring hierarchies through customized interactive mapping activities. These activities focus on interactive web-based Geographic Information System (GIS) and are relevant to today’s problems and challenges. Written in a guided, hands-on, understandable manner, all activities are designed to build practical and problem-solving skills that rest on mathematical principles and move students from thinking about maps as references that focus solely on where is something, to analytical tools, focusing primarily on the whys of where. Success with this transition through interaction permits most readers to master mathematical concepts and GIS tools. FEATURES Offers custom-designed geographical activities to fit with specific mathematical topics Helps students become comfortable using mathematics in a variety of professions Provides an innovative, engaging, and practical set of activities to ease readers through typically difficult, often elementary, mathematical topics: fractions, the distributive law, and much more Uses web-based GIS maps, apps, and other tools and data that can be accessed on any device, anywhere, at any time, requiring no prior GIS background Written by experienced teachers and researchers with lifelong experience in teaching mathematics, geography, and spatial analysis This textbook applies to undergraduate and graduate students in universities and community colleges including those in basic mathematics courses, as well as upper-level undergraduate and graduate students taking courses in geographic information systems, remote sensing, photogrammetry, geography, geodesy, information science, engineering, and geology. Professionals interested in learning techniques and technologies for collecting, analyzing, managing, processing, and visualizing geospatial datasets will also benefit from this book as they refresh their knowledge in mathematics. |
| a train leaves math problem: Mission to Mars Eric Walters, 2023-04-11 Teen astronaut Houston Williams is now the pilot on a mission to Mars. Houston Williams was thrilled to win a scholarship to attend a space program at NASA. What he didn’t realize was that organizers were recruiting people for a top-secret research project aimed at studying how space travel affects people of different ages. After months spent on the International Space Station conducting a variety of experiments, Houston is surprised to learn that he and his two friends Ashley and Teal have been chosen to join a highly political mission to Mars. But after tragedy strikes, the teens are forced to continue the mission on their own. This is the third book in the Teen Astronauts series, following Houston, Is There A Problem? and Boldly Go. |
| a train leaves math problem: Fostering Children's Mathematical Power Arthur J. Baroody, Ronald T. Coslick, 1998-09-01 Teachers have the responsibility of helping all of their students construct the disposition and knowledge needed to live successfully in a complex and rapidly changing world. To meet the challenges of the 21st century, students will especially need mathematical power: a positive disposition toward mathematics (curiosity and self confidence), facility with the processes of mathematical inquiry (problem solving, reasoning and communicating), and well connected mathematical knowledge (an understanding of mathematical concepts, procedures and formulas). This guide seeks to help teachers achieve the capability to foster children's mathematical power - the ability to excite them about mathematics, help them see that it makes sense, and enable them to harness its might for solving everyday and extraordinary problems. The investigative approach attempts to foster mathematical power by making mathematics instruction process-based, understandable or relevant to the everyday life of students. Past efforts to reform mathematics instruction have focused on only one or two of these aims, whereas the investigative approach accomplishes all three. By teaching content in a purposeful context, an inquiry-based fashion, and a meaningful manner, this approach promotes chilren's mathematical learning in an interesting, thought-provoking and comprehensible way. This teaching guide is designed to help teachers appreciate the need for the investigative approach and to provide practical advice on how to make this approach happen in the classroom. It not only dispenses information, but also serves as a catalyst for exploring, conjecturing about, discussing and contemplating the teaching and learning of mathematics. |
| a train leaves math problem: Navigating MathLand Linda Kasal Fusco, 2017-06-21 Navigating MathLand uses a unique lens to focus on how students prefer to learn mathematics. The intent of this book is to provide a guide for parents to help them navigate the thirteen years of their children’s math education (K-12). The book will provide parents with the knowledge and skills they will need to proactively advocate for their children’s preparation for the 21st century workforce. |
| a train leaves math problem: Something Else Entirely Thomas M Hall, 2007-05 Benny Curtis is a twelve-year-old boy with an eidetic memory-the ability to remember almost everything he sees or hears. Benny's grandfather is a retired humanities professor who loves everything historical. So when he starts spending a lot of time with his grandfather, Benny's mind becomes filled with the history of all sorts of subjects-baseball, movies, art, TV, rock and roll, etc. With this historical context and his incredible memory, Benny is able to view the world in a unique way. Last year when Benny was eleven, a number of amazing things occurred, not supernatural amazing, but real-life amazing. But in order for you to appreciate these remarkable events, you need to hear what led up to them. Benny's journey is filled with humorous observations, anecdotes, and reflections, as he analyzes the world around him. |
| a train leaves math problem: Math William C. Robertson, 2006 Flummoxed by formulas? Queasy about equations? Perturbed by pi? Now you can stop cursing over calculus and start cackling over Math, the newest volume in Bill RobertsonOCOs accurate but amusing Stop Faking It best sellers. As Robertson sees it, too many people view mathematics as a set of rules to be followed, procedures to memorize, and theorems to apply. This book focuses on the reasoning behind the rules, from math basics all the way up to a brief introduction to calculus. |
| a train leaves math problem: Oswaal CTET (CENTRAL TEACHERS ELIGIBILITY TEST) 15 previous years Solved papers PAPER - I (Classes 1 to 5) YEAR-WISE (2013 - 2023) For 2024 Exam , 2023-10-19 Description of the Product: 1. 100% Updated with latest fully solved papers of Sept. 2023 2. Extensive Practice with 2200+ No. of Questions in Each Subject 3. Crisp Revision with Smart Mind Maps 4. Valuable Exam Insights with Expert Tips to crack CTET in first attempt 5. Concept Clarity with 15 solved papers (2013 to 2023) with Detailed Explanations 6. 100% Exam Readiness with 5 Years Chapter-wise Trend Analysis (2019-2023) |
| a train leaves math problem: Too Smart Jones and the Stolen Bicycles Gilbert Morris, 2000 Juliet Jones swings into action when bicycles begin to disappear throughout the neighborhood. |
| a train leaves math problem: Macmillan/McGraw-Hill Math: Teacher ed., v. 1 , 2004 |
| a train leaves math problem: Wonderlic Test Strategy! Winning Multiple Choice Strategies for increasing your score on the Wonderlic! Complete Test Preparation Inc., 2020-07-27 Learn and Practice Proven multiple choice strategies for Reading Comprehension, Word Problems and Basic Math! If you are preparing for the WONDERLIC®, you probably want all the help you can get! WONDERLIC® Test Strategy is your complete guide to answering multiple choice questions! You will learn: - Powerful multiple choice strategies with practice questions for each strategy. Learn 14 powerful multiple choice strategies and then practice. Answer key for all practice questions with extensive commentary including tips, short-cuts and strategies. - How to prepare for a multiple choice exam - make sure you are preparing properly and not wasting valuable study time! - Who does well on multiple choice exams and who does not - and how to make sure you do! - How to handle trick questions - usually there are one or two trick questions to separate the really good students from the rest - tips and strategies to handle these special questions. - Math short-cuts, tips and tricks that will save you valuable exam time! - Step-by-step strategy for answering multiple choice - on any subject! - Common Mistakes on a Test - and how to avoid them - How to avoid test anxiety - how to avoid one of the most common reasons for low scores on a test - How to prepare for a test - proper preparation for your exam will definitely boost your score! - How to psych yourself up for a test - tips on the all-important mental preparation! - Learn what you must do in the test room Multiple choice strategies and practice questions for basic math, reading comprehension and word problems. Wonderlic is a registered trademark of Wonderlic Personnel Test, Inc., who are not involved in the production of, and do not endorse this product. Includes over 150 practice questions! Once you learn our powerful multiple choice strategy techniques, practice them right away on reading comprehension, basic math and word problems! Also included is How to Take a Test - The Complete Guide Let’s face it: test-taking is really not easy! While some people seem to have the natural ability to know what to study, how to absorb and retain information, and how to stay calm enough while actually taking a test to earn a great score, most of us find taking tests to be sheer misery. This is one of the most important chapters! Here you will find out: How to Take a Test - The basics In the Test Room - What you MUST do Common Mistakes on a Test - And how to avoid them Mental Prep - How to psych yourself up for a test Maybe you have read this kind of thing before, and maybe feel you don't need it, and you are not sure if you are going to buy this book. Remember though, it only a few percentage points divide the PASS from the FAIL students. Even if our test tips increase your score by a few percentage points, isn't that worth it? Remember it only a few percentage points divide the PASS from the FAIL students. Why not do everything you can to increase your score? |
| a train leaves math problem: Automatic Documentation and Mathematical Linguistics , 1995 |
www.train.org
TRAIN is a national learning network providing training opportunities and resources for public health professionals.
www.train.org
TRAIN is a national learning network providing training opportunities and resources for public health professionals.
