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Integer Word Problems: Mastering the Art of Problem Solving
Are you staring at a word problem involving integers, feeling a surge of anxiety? Don't worry, you're not alone! Many students find integer word problems challenging, but with the right approach and understanding, they can become surprisingly straightforward. This comprehensive guide will equip you with the skills and strategies to tackle integer word problems with confidence. We'll break down the process step-by-step, providing clear examples and techniques to help you master this essential math skill. Get ready to conquer those integer word problems!
Understanding Integers: The Foundation
Before diving into word problems, let's refresh our understanding of integers. Integers are whole numbers (no fractions or decimals) that can be positive, negative, or zero. Understanding the relationships between positive and negative integers – especially addition, subtraction, multiplication, and division – is crucial. Remember the rules of signs:
Adding: Adding two positive integers results in a positive integer. Adding two negative integers results in a negative integer. Adding a positive and a negative integer requires finding the difference and using the sign of the larger number.
Subtracting: Subtracting an integer is the same as adding its opposite. For example, 5 – (-3) = 5 + 3 = 8.
Multiplying/Dividing: Multiplying or dividing two integers with the same sign results in a positive integer. Multiplying or dividing two integers with different signs results in a negative integer.
Deconstructing Integer Word Problems: A Step-by-Step Approach
Tackling integer word problems effectively requires a systematic approach. Here's a proven method:
#### 1. Read Carefully and Identify Key Information:
The first step is to carefully read the problem multiple times. Underline or highlight key information, such as numbers, keywords (increase, decrease, profit, loss, above, below, etc.), and the question being asked.
#### 2. Define Variables:
Assign variables (usually letters like x, y, z) to represent the unknown quantities in the problem. Clearly define what each variable represents. For example, "Let x represent the temperature increase."
#### 3. Translate Words into Mathematical Equations:
This is often the trickiest part. Carefully translate the words into mathematical expressions. Look for keywords that indicate mathematical operations:
"Sum," "total," "more than," "increased by" suggest addition (+).
"Difference," "less than," "decreased by," "minus" suggest subtraction (-).
"Product," "times," "multiplied by" suggest multiplication (×).
"Quotient," "divided by" suggest division (÷).
#### 4. Solve the Equation:
Use your algebraic skills to solve the equation you've created. Remember to show your work clearly, step-by-step.
#### 5. Check Your Answer:
Finally, check your answer to ensure it makes sense in the context of the word problem. Does the answer logically address the question asked?
Examples of Integer Word Problems
Let's illustrate with a few examples:
Example 1: The temperature in the morning was -5°C. By afternoon, it increased by 8°C. What was the afternoon temperature?
Variables: Let x = afternoon temperature.
Equation: x = -5 + 8
Solution: x = 3°C
Example 2: A submarine dives 250 meters below sea level, then ascends 100 meters. What is its final depth?
Variables: Let x = final depth.
Equation: x = -250 + 100
Solution: x = -150 meters (150 meters below sea level)
Example 3: A company made a profit of $5,000 in the first quarter and a loss of $2,000 in the second quarter. What was their net profit or loss for the two quarters?
Variables: Let x = net profit/loss.
Equation: x = 5000 + (-2000)
Solution: x = $3,000 (net profit)
Advanced Integer Word Problems: Consecutive Integers and More
More complex problems might involve consecutive integers (integers that follow one another, like 3, 4, 5) or other more intricate relationships. These problems often require setting up systems of equations. Practice is key to mastering these advanced problems.
Conclusion
Mastering integer word problems takes practice and a systematic approach. By following the steps outlined above – carefully reading, defining variables, translating words into equations, solving, and checking – you can build confidence and significantly improve your problem-solving skills. Remember, every problem is a learning opportunity. Don't be discouraged by initial challenges; keep practicing, and you'll see your abilities grow.
FAQs
1. What are some common keywords to look for in integer word problems? Look for words like "sum," "difference," "product," "quotient," "increase," "decrease," "profit," "loss," "above," "below," "more than," "less than."
2. How can I improve my speed in solving integer word problems? Practice consistently, focusing on understanding the underlying concepts and developing efficient problem-solving strategies.
3. What resources are available to help me practice integer word problems? Many online resources, textbooks, and practice workbooks offer a wide range of integer word problems with varying difficulty levels.
4. What if I get stuck on a problem? Break the problem down into smaller parts, reread the problem carefully, and try different approaches. If you're still stuck, seek help from a teacher, tutor, or online forum.
5. Are there any shortcuts or tricks for solving integer word problems? While there aren't magical shortcuts, understanding the properties of integers and practicing regularly will significantly improve your efficiency and accuracy.
| integer word problems: CliffsQuickReview Math Word Problems Karen L. Anglin, 2007-05-21 CliffsQuickReview course guides cover the essentials of your toughest classes. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions. CliffsQuickReview Math Word Problems gives you a clear, concise, easy-to-use review of the basics of solving math word problems. Introducing each topic, defining key terms, and carefully walking you through each sample problem gives you insight and understanding to solving math word problems. You begin by building a strong foundation in translating expressions, inserting parentheses, and simplifying expressions. On top of that base, you can build your skills for solving word problems: Discover the six basic steps for solving word problems Translate English-language statements into equations and then solve them Solve geometry problems involving single and multiple shapes Work on proportion and percent problems Solve summation problems by using the Board Method Use tried-and-true methods to solve problems about money, investments, mixtures, and distance CliffsQuickReview Math Word Problems acts as a supplement to your textbook and to classroom lectures. Use this reference in any way that fits your personal style for study and review — you decide what works best with your needs. Here are just a few ways you can search for information: View the chapter on common errors and how to avoid them Get a glimpse of what you’ll gain from a chapter by reading through the Chapter Check-In at the beginning of each chapter Use the Chapter Checkout at the end of each chapter to gauge your grasp of the important information you need to know Test your knowledge more completely in the CQR Review and look for additional sources of information in the CQR Resource Center Use the glossary to find key terms fast With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades. |
| integer word problems: Word Problems, Grade 7 , 2013-12-02 Spectrum(R) Word Problems for grade 7 includes practice for essential math skills, such as real world applications, multi-step word problems, variables, ratio and proportion, perimeter, area and volume, percents, statistics and more. Spectrum(R) Word Problems supplement to classroom work and proficiency test preparation. The series provides examples of how the math skills students learn in school apply to everyday life with challenging, multi-step word problems. It features practice with word problems that are an essential part of the Common Core State Standards. Word problem practice is provided for essential math skills, such as fractions, decimals, percents, metric and customary measurement, graphs and probability, and preparing for algebra and more. |
| integer word problems: Algebra Word Problems Rebecca Wingard-Nelson, 2013-09 Having a problem with word problems? Author Rebecca Wingard-Nelson introduces simple ways to tackle tricky word problems with algebra. Real world examples make the book easy to read and are great for students to use on their own, or with parents, teachers, or tutors. Free downloadable worksheets are available on www.enslow.com. |
| integer word problems: Math Word Problems For Dummies Mary Jane Sterling, 2008-02-05 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 |
| integer word problems: Exploring the Integer Addition and Subtraction Landscape Laura Bofferding, Nicole M. Wessman-Enzinger, 2018-08-24 Over the past few decades there has been increased interest in how students and teachers think and learn about negative numbers from a variety of perspectives. In particular, there has been debate about when integers should be taught and how to teach them to best support students’ learning. This book brings together recent work from researchers to illuminate the state of our understanding about issues related to integer addition and subtraction with a goal of highlighting how the variety of perspectives support each other or contribute to the field in unique ways. In particular, this book focuses on three main areas of integer work: students’ thinking, models and metaphors, and teachers’ thinking. Each chapter highlights a theoretically guided study centered on integer addition and subtraction. Internationally known scholars help connect the perspectives and offer additional insights through section commentaries. This book is an invaluable resource to those who are interested in mathematics education and numerical thinking. |
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| integer word problems: The Bounded and Precise Word Problems for Presentations of Groups S. V. Ivanov, 2020-05-13 The author introduces and studies the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE, i.e., it can be solved in polynomial space. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, the author obtains polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. The author also obtains polynomial time bounds for these problems. |
| integer word problems: Learning Mathematics Anthony Orton, 2004-10-01 • Why do some students achieve more than others? • Do we have to wait until pupils are ready? • Can children discover math for themselves? • Does language interfere with the learning of math? This classic text, written from the viewpoint of the math teacher, provides answers to these and many more questions. Each chapter explores a particular issue that illustrates the interaction between theory and practice. New chapters have been included on cognition, pattern, and ICT. |
| integer word problems: Differentiating Instruction With Menus for the Inclusive Classroom Laurie E. Westphal, 2021-09-03 Differentiating Instruction With Menus for the Inclusive Classroom: Math for grades 6-8 offers teachers who have multiple ability levels in one classroom everything they need to create a student-centered learning environment based on choice. For each topic covered, there are two menus that look similar but contain differentiated content: one menu for students working on grade level and the other for students working below grade level. Using the creative, challenging choices found in Tic-Tac-Toe menus, List menus, 2-5-8 menus, and Game Show menus, students will demonstrate their knowledge with unique, exciting products. Also included are specific guidelines for products, assessment rubrics, and teacher introduction pages for each menu. These menus can also be used in conjunction with the Differentiating Instruction With Menus series (for students working above grade level) for three tiers of complementary menus. Grades 6-8 |
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| integer word problems: ICSE NumbersWiz Class 7 ANUBHUTI GANGAL, ICSE NumbersWiz is a series of books for KG to Class 8 which conforms to the latest CISCE curriculum. The main aim of writing this series is to help the children understand difficult mathematical concepts in a simple manner in easy language. |
| integer word problems: New Mathematics Today Class 7 ANUBHUTI GANGAL, New Mathematics Today, a thoroughly revised series for KG to Class 8, has been designed as per the requirements of the latest curriculum. The content of this series is designed to reach all learners in the classroom irrespective of their skill levels or learning capabilities. |
| integer word problems: Learning and Teaching Mathematics Peter Bryant, Terezinha Nunes, 2016-01-28 The authors of this volume, which is newly available in paperback, all hold the view that mathematics is a form of intelligent problem solving which plays an important part in children's lives outside the classroom as well as in it. Learning and Teaching Mathematics provides an exciting account of recent and radically different research on teaching and learning mathematics which will have a far reaching effect on views about mathematical education. |
| integer word problems: Word Problem Workbook: Student's Workbook Hammer, 2003-10 |
| integer word problems: Foundation Mathematics for Class 7 R. S. Aggarwal, Goyal Brothers Prakashan, 2019-01-01 The revised edition of the series Foundation Mathematics for Classes 6, 7 and 8 is based on the latest curriculum prepared and recommended by the Council for the Indian School Certificate Examinations, New Delhi. The present mathematics curriculum aims to develop a number ofMathematical Skills (like Numerical Calculation, Algebraic Manipulation, Spatial Visualisation, Data Analysis, Measurement, Estimation and Approximation) and Mathematical Processes (like Reasoning, Communication and Connections, Problem solving and Heuristics, Estimation, Technology etc.) among students at these levels. This series has been developed and designed keeping in mind the following objectives of the latest curriculum : Students should : • Enjoy learning of mathematics. • Learn important mathematics that is much more than few formulas and mechanical procedures of solving problems. • Pose and solve meaningful problems. • See mathematics as something to talk about, to communicate, to discuss among themselves, to work together on. • Understand the basic structure of mathematics : Arithmetic, algebra, geometry and trigonometry, the basic content areas of school mathematics, all offer a methodology of abstraction, structuration and generalization. Goyal Brothers Prakashan |
| integer word problems: Algebraic Equations Speedy Publishing, 2014-09-23 One exceptionally important part of knowing how to do algebra is knowing all of the equations. This can be very difficult for a student who is first learning how to do algebra, and it can be very difficult for one to sit down to learn them. One great way for persons to avoid simply starring at the equation for hours is to use an algebra equation study guide. The guide contains useful exercises a student can do in order to learn the problems. |
| integer word problems: Rigor for Students with Special Needs Barbara R. Blackburn, Bradley S. Witzel, 2021-09-27 This practical guide explains how to raise the rigor for students with special needs so they can achieve higher levels of learning. Bestselling author Barbara R. Blackburn and intervention expert Bradley S. Witzel provide helpful information on assessment, planning, co-teaching models, high expectations, common obstacles, and emphasizing positive outcomes. This second edition offers new, expanded instructional strategies for literacy and math, as well as strategies that work across subject areas. In addition, each chapter is filled with tools and examples to help you implement the ideas. Perfect for general and special educators and supervisors, the book also comes with a study guide so you can collaborate on the book with building or district colleagues. With the practical information in this book, you’ll understand how to teach with higher expectations and rigor so that all students can feel successful. |
| integer word problems: Methods of Solving Nonstandard Problems Ellina Grigorieva, 2015-09-17 This book, written by an accomplished female mathematician, is the second to explore nonstandard mathematical problems – those that are not directly solved by standard mathematical methods but instead rely on insight and the synthesis of a variety of mathematical ideas. It promotes mental activity as well as greater mathematical skills, and is an ideal resource for successful preparation for the mathematics Olympiad. Numerous strategies and techniques are presented that can be used to solve intriguing and challenging problems of the type often found in competitions. The author uses a friendly, non-intimidating approach to emphasize connections between different fields of mathematics and often proposes several different ways to attack the same problem. Topics covered include functions and their properties, polynomials, trigonometric and transcendental equations and inequalities, optimization, differential equations, nonlinear systems, and word problems. Over 360 problems are included with hints, answers, and detailed solutions. Methods of Solving Nonstandard Problems will interest high school and college students, whether they are preparing for a math competition or looking to improve their mathematical skills, as well as anyone who enjoys an intellectual challenge and has a special love for mathematics. Teachers and college professors will be able to use it as an extra resource in the classroom to augment a conventional course of instruction in order to stimulate abstract thinking and inspire original thought. |
| integer word problems: Posing and Solving Mathematical Problems Patricio Felmer, Erkki Pehkonen, Jeremy Kilpatrick, 2016-04-29 This book collects recent research on posing and solving mathematical problems. Rather than treating these two crucial aspects of school mathematics as separate areas of study, the authors approach them as a unit where both areas are measured on equal grounds in relation to each other. The contributors are from a vast variety of countries and with a wide range of experience; it includes the work from many of the leading researchers in the area and an important number of young researchers. The book is divided in three parts, one directed to new research perspectives and the other two directed to teachers and students, respectively. |
| integer word problems: Intermediate Algebra Martin M. Zuckerman, 1986-01-01 As the title implies, this is a straightforward approach to intermediate algebra that guides the student step-by-step toward acquiring mathematical skills with examples, exercises, and more examples and exercises. The book is first and foremost student oriented. It does not dwell on formalities but appeals to intuition. Geometric arguments are favored wherever possible. Textual explanations are precise, brief, and to the point and are always accompanied by illustrative examples. |
| integer word problems: Praxis Core For Dummies, with Online Practice Tests Carla C. Kirkland, Chan Cleveland, 2014-09-02 As a future educator, you know how thorough preparation can affect performance, and the Praxis Core Academic Skills for Educators exam requires your very best. This guide features the most updated information, and provides a detailed overview so you know what to expect on test day. |
| integer word problems: Proceedings of the Nineteenth Annual Conference of the Cognitive Science Society Michael G. Shafto, Pat Langley, 1997 This volume features the complete text of the material presented at the Nineteenth Annual Conference of the Cognitive Science Society. Papers have been loosely grouped by topic and an author index is provided in the back. As in previous years, the symposium included an interesting mixture of papers on many topics from researchers with diverse backgrounds and different goals, presenting a multifaceted view of cognitive science. In hopes of facilitating searches of this work, an electronic index on the Internet's World Wide Web is provided. Titles, authors, and summaries of all the papers published here have been placed in an online database which may be freely searched by anyone. You can reach the web site at: www-csli.stanford.edu/cogsci97. |
| integer word problems: Linear-Fractional Programming Theory, Methods, Applications and Software E.B. Bajalinov, 2013-12-01 This is a book on Linear-Fractional Programming (here and in what follows we will refer to it as LFP). The field of LFP, largely developed by Hungarian mathematician B. Martos and his associates in the 1960's, is concerned with problems of op timization. LFP problems deal with determining the best possible allo cation of available resources to meet certain specifications. In particular, they may deal with situations where a number of resources, such as people, materials, machines, and land, are available and are to be combined to yield several products. In linear-fractional programming, the goal is to determine a per missible allocation of resources that will maximize or minimize some specific showing, such as profit gained per unit of cost, or cost of unit of product produced, etc. Strictly speaking, linear-fractional programming is a special case of the broader field of Mathematical Programming. LFP deals with that class of mathematical programming problems in which the relations among the variables are linear: the con straint relations (i.e. the restrictions) must be in linear form and the function to be optimized (i.e. the objective function) must be a ratio of two linear functions. |
| integer word problems: Mathematics Teaching, Learning, and Liberation in the Lives of Black Children Danny Bernard Martin, 2010-06-21 With issues of equity at the forefront of mathematics education research and policy, this collection offers authoritative scholarship that sheds light on the ways that young black learners experience mathematics in schools and their communities. |
| integer word problems: Scholar's New Syllabus Composite Mathematics 6 V.K.Kapoor, |
| integer word problems: Homework Helpers: Basic Math and Pre-Algebra, Revised Edition Denise Szecsei, 2011-04-15 Homework Helpers: Basic Math and Pre-Algebrawill help build a solid mathematical foundation and enable students to gain the confidence they need to continue their education in mathematics. Particular attention is placed on topics that students traditionally struggle with the most. The topics are explained in everyday language before the examples are worked. The problems are solved clearly and systematically, with step-by-step instructions provided. Problem-solving skills and good habits, such as checking your answers after every problem, are emphasized along with practice problems throughout, and the answers to all of the practice problems are provided. Homework Helpers: Basic Math and Pre-Algebra is a straightforward and easy-to-read review of arithmetic skills. It includes topics that are intended to help prepare students to successfully learn algebra, including: Working with fractions Understanding the decimal system Calculating percentages Solving linear equalities Graphing functions Understanding word problems |
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| integer word problems: Intermediate Algebra with Trigonometry Charles P. McKeague, 2014-05-10 Intermediate Algebra with Trigonometry focuses on principles, operations, and approaches employed in intermediate algebra with trigonometry. The publication first elaborates on basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on polynomials, sums, and differences, multiplication of polynomials, greatest common factor and factoring by grouping, inequalities involving absolute value, equations with absolute value, and multiplication, division, and order of operation for real numbers. The manuscript then ponders on rational expressions, quadratic equations, and rational expressions and roots. Topics include equations quadratic in form, quadratic formula, completing the square, multiplication and division of complex numbers, equations with radicals, simplified form for radicals, multiplication and division of rational expressions, and addition and subtraction of rational expressions. The text takes a look at triangles, trigonometric identities and equations, introduction to trigonometry, and sequence and series, including arithmetic progressions, trigonometric functions, tables and calculators, sum and difference formulas, and the law of sines and cosines. The publication is a valuable reference for students and researchers interested in intermediate algebra with trigonometry. |
| integer word problems: New GRE, Miller's Math Bob Miller, 2013-01-01 If you’re worried about your math score on the new GRE, get the guide that teaches you everything you need to know! Bob Miller’s Math for the New GRE Gets You into Grad School! Bob Miller has taught math to thousands of students at all educational levels for 30 years. His proven teaching methods help grad-school-bound students succeed on the math portion of the new GRE General Test. Written in a lively and unique format, Bob Miller’s Math for the New GRE is fully aligned with the new GRE General Test that launched in August 2011. This book is the perfect study companion for anyone taking the new GRE General. Bob Miller addresses the changes to the content and format of the exam while teaching math in an easy-to-understand style. Unlike some dull test preps that merely present the material, Bob actually teaches and explains math concepts and ideas. His no-nonsense, no-stress teaching methods and decades of experience as a math teacher help you master the material and get an excellent score. The book contains everything GRE test-takers need to know to solve the math problems that typify the Quantitative section of the exam. Each chapter is devoted to a specific topic and is packed with examples and exercises that reinforce the required math skills. Bob Miller’s Math for the New GRE is a must-have for anyone who needs to boost their math skills before taking the new GRE! |
| integer word problems: Problem Solving & Comprehension Arthur Whimbey, Jack Lochhead, Ron Narode, 2013-06-26 This popular book shows students how to increase their power to analyze problems and comprehend what they read using the Think Aloud Pair Problem Solving [TAPPS] method. First it outlines and illustrates the method that good problem solvers use in attacking complex ideas. Then it provides practice in applying this method to a variety of comprehension and reasoning questions, presented in easy-to-follow steps. As students work through the book they will see a steady improvement in their analytical thinking skills and become smarter, more effective, and more confident problem solvers. Not only can using the TAPPS method assist students in achieving higher scores on tests commonly used for college and job selection, it teaches that problem solving can be fun and social, and that intelligence can be taught. Changes in the Seventh Edition: New chapter on open-ended problem solving that includes inductive and deductive reasoning; extended recommendations to teachers, parents, and tutors about how to use TAPPS instructionally; Companion Website with PowerPoint slides, reading lists with links, and additional problems. |
| integer word problems: Intermediate Algebra Charles P. McKeague, 2014-05-10 Intermediate Algebra focuses on the principles, operations, and approaches involved in intermediate algebra. The book first elaborates on basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on the greatest common factor and factoring by grouping, factoring trinomials, special factoring, equations with absolute value, inequalities involving absolute value, formulas, first-degree equations, graphing simple and compound inequalities, and properties of real numbers. The text then takes a look at rational expressions, rational exponents and roots, and quadratic equations. Topics include solving quadratic equations by factoring, discriminant and the sum and product of solutions, multiplication and division of complex numbers, combinations of radical expressions, rational exponents, complex fractions, and multiplication and division of rational expressions. The manuscript elaborates on sequence and series, logarithms, relations and functions, and conic sections, including ellipses and hyperbolas, nonlinear systems, function and notation, algebra with functions, common logarithms and computations, and word problems. The publication is a dependable reference for students and researchers interested in intermediate algebra. |
| integer word problems: Bridging the Gap Between Arithmetic & Algebra Bradley S. Witzel, 2015-11-15 Although two federal panels have concluded that all students can learn mathematics and most can succeed through Algebra 2, the abstractness of algebra and missing precursor understandings may be overwhelming to many students … and their teachers. Bridging the Gap Between Arithmetic & Algebra responds to this need for instruction and interventions that go beyond typical math lesson plans. Providing a review of evidence-based practices, the book is an essential reference for mathematics teachers and special education teachers when teaching mathematics to students who struggle with the critical concepts and skills necessary for success in algebra. Audiences: General education (mathematics) teachers, special education teachers, administrators, teacher educators. |
| integer word problems: Differentiating Instruction With Menus Laurie E. Westphal, 2021-09-03 The best-selling Differentiating Instruction With Menus series has helped teachers nationwide differentiate instruction for their high-ability learners with easy-to-use menus and exciting tools to challenge and reach gifted and advanced students in the classroom. Each book includes an updated, student-friendly rubric that can assess different types of products, free choice proposal forms to encourage independent study, and new and favorite challenging menus to meet the needs of these diverse higher level learners. Readers will also be able to save time by using updated guidelines that reflect changes in technology for each of the products included in the menus and find direct alignment with standards approved in recent years. Topics addressed in Differentiating Instruction With Menus: Math (Grades 6-8, 2nd ed.) include numbers and operations, geometry, measurement, and basic algebra. Grades 6-8 |
| integer word problems: Arithmetic Functions and Integer Products P.D.T.A. Elliott, 2012-12-06 Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = ± I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be non negative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x». Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic func tions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239/240 of this series, under the title Probabilistic Number Theory. |
| integer word problems: Math Word Problems Demystified 2/E Allan Bluman, 2011-07-19 Your solution to MATH word PROBLEMS! Find yourself stuck on the tracks when two trains are traveling at different speeds? Help has arrived! Math Word Problems Demystified, Second Edition is your ticket to problem-solving success. Based on mathematician George Polya's proven four-step process, this practical guide helps you master the basic procedures and develop a plan of action you can use to solve many different types of word problems. Tips for using systems of equations and quadratic equations are included. Detailed examples and concise explanations make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning. It's a no-brainer! You'll learn to solve: Decimal, fraction, and percent problems Proportion and formula problems Number and digit problems Distance and mixture problems Finance, lever, and work problems Geometry, probability, and statistics problems Simple enough for a beginner, but challenging enough for an advanced student, Math Word Problems Demystified, Second Edition helps you master this essential mathematics skill. |
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| integer word problems: WORD PROBLEMS II Lev D. Beklemishev, 2000-04-01 WORD PROBLEMS II |
| integer word problems: Differentiated Instruction for the Middle School Math Teacher Karen E. D'Amico, Kate Gallaway, 2008-01-02 Differentiated Instruction for the Middle School Math Teacher is a practical and easy-to-use resource for teaching a standards-based math curriculum to all learners. It gives you effective ways to present math concepts, shows how to provide opportunities for guided practice, and offers ideas for modifying the material to provide access to the same content standard for all students in the inclusive classroom. This book also contains key strategies for collaborating with other professionals, suggestions for involving the students' families by tying math concepts to students' everyday lives, and valuable assessment strategies. The lessons in the book cover middle school math topics correlated to the standards of the National Council of Teachers of Math, ranging from numbers and operations to problem solving and reasoning. Each lesson includes: Instructions for presenting the lesson to the whole class Worksheets designed to help review and reinforce theconcepts presented in each lesson A section on how to adapt the lesson for the inclusive classroom, including descriptions of different stations for different learners A home-school connection with family-based everyday math activities Suggestions for how to assess students' grasp of the concepts presented in the lesson |
| integer word problems: Alpha Teach Yourself Algebra I in 24 Hours Jane Cook, 2011-01-04 The first step in complex math is now the easiest. Alpha Teach Yourself Algebra I in 24 Hours provides readers with a structured, self-paced, straight-forward tutorial to algebra. It's the perfect textbook companion for students struggling with algebra, a solid primer for those looking to get a head start on an upcoming class, and a welcome refresher for parents tasked with helping out with homework, all in 24 one-hour lessons. • Algebra is the second-most popular mathematic course for college- bound high school students • Nearly all college-bound high school students now take algebra |
| integer word problems: Constructing Number Anderson Norton, Martha W. Alibali, 2018-12-17 The book synergizes research on number across two disciplines—mathematics education and psychology. The underlying problem the book addresses is how the brain constructs number. The opening chapter frames the problem in terms of children’s activity, including mental and physical actions. Subsequent chapters are organized into sections that address specific domains of number: natural numbers, fractions, and integers. Chapters within each section address ways that children build upon biological primitives (e.g., subitizing) and prior constructs (e.g., counting sequences) to construct number. The book relies on co-authored chapters and commentaries at the end of each section to create dialogue between junior faculty and senior researchers, as well as between psychologists and mathematics educators. The final chapter brings this work together around the framework of children’s activity and additional themes that arise in the collective work. The book is aimed to appeal to mathematics educators, mathematics teacher educators, mathematics education researchers, educational psychologists, cognitive psychologists, and developmental psychologists. |
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