Elements of Mathematics for Economics and Finance

Size: px
Start display at page:

Download "Elements of Mathematics for Economics and Finance"

Transcription

1 Elements of Mathematics for Economics and Finance

2 Vassilis C. Mavron and Timothy N. Phillips Elements of Mathematics for Economics and Finance With 77 Figures

3 Vassilis C. Mavron, MA, MSc, PhD Institute of Mathematical and Physical Sciences University of Wales Aberystwyth Aberystwyth SY23 3BZ Wales, UK Timothy N. Phillips, MA, MSc, DPhil, DSc Cardiff School of Mathematics Cardiff University Senghennydd Road Cardiff CF24 4AG Wales, UK Mathematics Subject Classification (2000): 91-01; 91B02 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: ISBN-10: e-isbn Printed on acid-free paper ISBN-13: Springer-Verlag London Limited 2007 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed in the United States of America (HAM) Springer Science + Business Media, LLC springer.com

4 Preface The mathematics contained in this book for students of economics and finance has, for many years, been given by the authors in two single-semester courses at the University of Wales Aberystwyth. These were mathematics courses in an economics setting, given by mathematicians based in the Department of Mathematics for students in the Faculty of Social Sciences or School of Management. The choice of subject matter and arrangement of material reflect this collaboration and are a result of the experience thus obtained. The majority of students to whom these courses were given were studying for degrees in economics or business administration and had not acquired any mathematical knowledge beyond pre-calculus mathematics, i.e., elementary algebra. Therefore, the first-semester course assumed little more than basic precalculus mathematics and was based on Chapters 1 7. This course led on to the more advanced second-semester course, which was also suitable for students who had already covered basic calculus. The second course contained at most one of the three Chapters 10, 12, and 13. In any particular year, their inclusion or exclusion would depend on the requirements of the economics or business studies degree syllabuses. An appendix on differentials has been included as an optional addition to an advanced course. The students taking these courses were chiefly interested in learning the mathematics that had applications to economics and were not primarily interested in theoretical aspects of the subject per se. The authors have not attempted to write an undergraduate text in economics but instead have written a text in mathematics to complement those in economics. The simplicity of a mathematical theory is sometimes lost or obfuscated by a dense covering of applications at too early a stage. For this reason, the aim of the authors has been to present the mathematics in its simplest form, highlighting threads of common mathematical theory in the various topics of v

5 vi Elements of Mathematics for Economics and Finance economics. Some knowledge of theory is necessary if correct use is to be made of the techniques; therefore, the authors have endeavoured to introduce some basic theory in the expectation and hope that this will improve understanding and incite a desire for a more thorough knowledge. Students who master the simpler cases of a theory will find it easier to go on to the more difficult cases when required. They will also be in a better position to understand and be in control of calculations done by hand or calculator and also to be able to visualise problems graphically or geometrically. It is still true that the best way to understand a technique thoroughly is through practice. Mathematical techniques are no exception, and for this reason the book illustrates theory through many examples and exercises. We are grateful to Noreen Davies and Joe Hill for invaluable help in preparing the manuscript of this book for publication. Above all, we are grateful to our wives, Nesta and Gill, and to our children, Nicholas and Christiana, and Rebecca, Christopher, and Emily, for their patience, support, and understanding: this book is dedicated to them. Vassilis C. Mavron Aberystwyth United Kingdom Timothy N. Phillips Cardiff United Kingdom March 2006

6 Contents 1. Essential Skills Introduction Numbers Addition and Subtraction Multiplication and Division Evaluation of Arithmetical Expressions Fractions Multiplication and Division Decimal Representation of Numbers Standard Form Percentages PowersandIndices Simplifying Algebraic Expressions Multiplying Brackets Factorization Linear Equations Introduction Solution of Linear Equations Solution of Simultaneous Linear Equations GraphsofLinearEquations Slope of a Straight Line BudgetLines Supply and Demand Analysis Multicommodity Markets vii

7 viii Contents 3. Quadratic Equations Introduction GraphsofQuadraticFunctions Quadratic Equations Applications to Economics Functions of a Single Variable Introduction Limits Polynomial Functions ReciprocalFunctions Inverse Functions The Exponential and Logarithmic Functions Introduction ExponentialFunctions Logarithmic Functions ReturnstoScaleofProductionFunctions Cobb-DouglasProductionFunctions Compounding of Interest Applications of the Exponential Function in Economic Modelling Differentiation Introduction RulesofDifferentiation ConstantFunctions LinearFunctions PowerFunctions SumsandDifferencesofFunctions ProductofFunctions QuotientofFunctions TheChainRule Exponential and Logarithmic Functions MarginalFunctionsinEconomics Marginal Revenue and Marginal Cost Marginal Propensities ApproximationtoMarginalFunctions HigherOrderDerivatives ProductionFunctions...129

8 Contents ix 7. Maxima and Minima Introduction Local Properties of Functions Increasing and Decreasing Functions ConcaveandConvexFunctions Local or Relative Extrema GlobalorAbsoluteExtrema PointsofInflection Optimization of Production Functions Optimization of Profit Functions OtherExamples Partial Differentiation Introduction FunctionsofTwoorMoreVariables Partial Derivatives Higher Order Partial Derivatives Partial Rate of Change TheChainRuleandTotalDerivatives Some Applications of Partial Derivatives ImplicitDifferentiation ElasticityofDemand Utility Production Graphical Representations Optimization Introduction UnconstrainedOptimization Constrained Optimization Substitution Method Lagrange Multipliers The Lagrange Multiplier λ:aninterpretation IsoCurves Matrices and Determinants Introduction Matrix Operations Scalar Multiplication Matrix Addition Matrix Multiplication Solutions of Linear Systems of Equations

9 x Contents 10.4 Cramer srule MoreDeterminants SpecialCases Integration Introduction RulesofIntegration DefiniteIntegrals DefiniteIntegration:AreaandSummation Producer s Surplus Consumer ssurplus Linear Difference Equations Introduction DifferenceEquations First Order Linear Difference Equations Stability TheCobwebModel Second Order Linear Difference Equations Complementary Solutions Particular Solutions Stability Differential Equations Introduction First Order Linear Differential Equations Stability Nonlinear First Order Differential Equations SeparationofVariables SecondOrderLinearDifferentialEquations TheHomogeneousCase TheGeneralCase Stability A. Differentials Index...309

Guide to Teaching Computer Science

Guide to Teaching Computer Science Guide to Teaching Computer Science Orit Hazzan Tami Lapidot Noa Ragonis Guide to Teaching Computer Science An Activity-Based Approach Dr. Orit Hazzan Associate Professor Technion - Israel Institute of

More information

Mathematics. Mathematics

Mathematics. Mathematics Mathematics Program Description Successful completion of this major will assure competence in mathematics through differential and integral calculus, providing an adequate background for employment in

More information

Math Techniques of Calculus I Penn State University Summer Session 2017

Math Techniques of Calculus I Penn State University Summer Session 2017 Math 110 - Techniques of Calculus I Penn State University Summer Session 2017 Instructor: Sergio Zamora Barrera Office: 018 McAllister Bldg E-mail: sxz38@psu.edu Office phone: 814-865-4291 Office Hours:

More information

Honors Mathematics. Introduction and Definition of Honors Mathematics

Honors Mathematics. Introduction and Definition of Honors Mathematics Honors Mathematics Introduction and Definition of Honors Mathematics Honors Mathematics courses are intended to be more challenging than standard courses and provide multiple opportunities for students

More information

Statewide Framework Document for:

Statewide Framework Document for: Statewide Framework Document for: 270301 Standards may be added to this document prior to submission, but may not be removed from the framework to meet state credit equivalency requirements. Performance

More information

Course Syllabus for Math

Course Syllabus for Math Course Syllabus for Math 1090-003 Instructor: Stefano Filipazzi Class Time: Mondays, Wednesdays and Fridays, 9.40 a.m. - 10.30 a.m. Class Place: LCB 225 Office hours: Wednesdays, 2.00 p.m. - 3.00 p.m.,

More information

Mathematics subject curriculum

Mathematics subject curriculum Mathematics subject curriculum Dette er ei omsetjing av den fastsette læreplanteksten. Læreplanen er fastsett på Nynorsk Established as a Regulation by the Ministry of Education and Research on 24 June

More information

AGS THE GREAT REVIEW GAME FOR PRE-ALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS

AGS THE GREAT REVIEW GAME FOR PRE-ALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS AGS THE GREAT REVIEW GAME FOR PRE-ALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS 1 CALIFORNIA CONTENT STANDARDS: Chapter 1 ALGEBRA AND WHOLE NUMBERS Algebra and Functions 1.4 Students use algebraic

More information

Algebra 1, Quarter 3, Unit 3.1. Line of Best Fit. Overview

Algebra 1, Quarter 3, Unit 3.1. Line of Best Fit. Overview Algebra 1, Quarter 3, Unit 3.1 Line of Best Fit Overview Number of instructional days 6 (1 day assessment) (1 day = 45 minutes) Content to be learned Analyze scatter plots and construct the line of best

More information

Math 098 Intermediate Algebra Spring 2018

Math 098 Intermediate Algebra Spring 2018 Math 098 Intermediate Algebra Spring 2018 Dept. of Mathematics Instructor's Name: Office Location: Office Hours: Office Phone: E-mail: MyMathLab Course ID: Course Description This course expands on the

More information

HOLMER GREEN SENIOR SCHOOL CURRICULUM INFORMATION

HOLMER GREEN SENIOR SCHOOL CURRICULUM INFORMATION HOLMER GREEN SENIOR SCHOOL CURRICULUM INFORMATION Subject: Mathematics Year Group: 7 Exam Board: (For years 10, 11, 12 and 13 only) Assessment requirements: Students will take 3 large assessments during

More information

Mathematics Assessment Plan

Mathematics Assessment Plan Mathematics Assessment Plan Mission Statement for Academic Unit: Georgia Perimeter College transforms the lives of our students to thrive in a global society. As a diverse, multi campus two year college,

More information

Advanced Grammar in Use

Advanced Grammar in Use Advanced Grammar in Use A self-study reference and practice book for advanced learners of English Third Edition with answers and CD-ROM cambridge university press cambridge, new york, melbourne, madrid,

More information

Lecture Notes on Mathematical Olympiad Courses

Lecture Notes on Mathematical Olympiad Courses Lecture Notes on Mathematical Olympiad Courses For Junior Section Vol. 2 Mathematical Olympiad Series ISSN: 1793-8570 Series Editors: Lee Peng Yee (Nanyang Technological University, Singapore) Xiong Bin

More information

Algebra 1 Summer Packet

Algebra 1 Summer Packet Algebra 1 Summer Packet Name: Solve each problem and place the answer on the line to the left of the problem. Adding Integers A. Steps if both numbers are positive. Example: 3 + 4 Step 1: Add the two numbers.

More information

IMPLEMENTING EUROPEAN UNION EDUCATION AND TRAINING POLICY

IMPLEMENTING EUROPEAN UNION EDUCATION AND TRAINING POLICY IMPLEMENTING EUROPEAN UNION EDUCATION AND TRAINING POLICY Implementing European Union Education and Training Policy A Comparative Study of Issues in Four Member States Edited by David Phillips Department

More information

GUIDE TO THE CUNY ASSESSMENT TESTS

GUIDE TO THE CUNY ASSESSMENT TESTS GUIDE TO THE CUNY ASSESSMENT TESTS IN MATHEMATICS Rev. 117.016110 Contents Welcome... 1 Contact Information...1 Programs Administered by the Office of Testing and Evaluation... 1 CUNY Skills Assessment:...1

More information

GCSE Mathematics B (Linear) Mark Scheme for November Component J567/04: Mathematics Paper 4 (Higher) General Certificate of Secondary Education

GCSE Mathematics B (Linear) Mark Scheme for November Component J567/04: Mathematics Paper 4 (Higher) General Certificate of Secondary Education GCSE Mathematics B (Linear) Component J567/04: Mathematics Paper 4 (Higher) General Certificate of Secondary Education Mark Scheme for November 2014 Oxford Cambridge and RSA Examinations OCR (Oxford Cambridge

More information

Math 150 Syllabus Course title and number MATH 150 Term Fall 2017 Class time and location INSTRUCTOR INFORMATION Name Erin K. Fry Phone number Department of Mathematics: 845-3261 e-mail address erinfry@tamu.edu

More information

Learning Disability Functional Capacity Evaluation. Dear Doctor,

Learning Disability Functional Capacity Evaluation. Dear Doctor, Dear Doctor, I have been asked to formulate a vocational opinion regarding NAME s employability in light of his/her learning disability. To assist me with this evaluation I would appreciate if you can

More information

THE PROMOTION OF SOCIAL AWARENESS

THE PROMOTION OF SOCIAL AWARENESS THE PROMOTION OF SOCIAL AWARENESS Powerful Lessons from the Partnership of Developmental Theory and Classroom Practice Robert L. Selman Russell Sage Foundation New York The Russell Sage Foundation The

More information

The Algebra in the Arithmetic Finding analogous tasks and structures in arithmetic that can be used throughout algebra

The Algebra in the Arithmetic Finding analogous tasks and structures in arithmetic that can be used throughout algebra Why Didn t My Teacher Show Me How to Do it that Way? Rich Rehberger Math Instructor Gallatin College Montana State University The Algebra in the Arithmetic Finding analogous tasks and structures in arithmetic

More information

Grade 6: Correlated to AGS Basic Math Skills

Grade 6: Correlated to AGS Basic Math Skills Grade 6: Correlated to AGS Basic Math Skills Grade 6: Standard 1 Number Sense Students compare and order positive and negative integers, decimals, fractions, and mixed numbers. They find multiples and

More information

The Indices Investigations Teacher s Notes

The Indices Investigations Teacher s Notes The Indices Investigations Teacher s Notes These activities are for students to use independently of the teacher to practise and develop number and algebra properties.. Number Framework domain and stage:

More information

Foothill College Summer 2016

Foothill College Summer 2016 Foothill College Summer 2016 Intermediate Algebra Math 105.04W CRN# 10135 5.0 units Instructor: Yvette Butterworth Text: None; Beoga.net material used Hours: Online Except Final Thurs, 8/4 3:30pm Phone:

More information

Answers To Hawkes Learning Systems Intermediate Algebra

Answers To Hawkes Learning Systems Intermediate Algebra Answers To Hawkes Learning Free PDF ebook Download: Answers To Download or Read Online ebook answers to hawkes learning systems intermediate algebra in PDF Format From The Best User Guide Database Double

More information

Instructor: Matthew Wickes Kilgore Office: ES 310

Instructor: Matthew Wickes Kilgore Office: ES 310 MATH 1314 College Algebra Syllabus Instructor: Matthew Wickes Kilgore Office: ES 310 Longview Office: LN 205C Email: mwickes@kilgore.edu Phone: 903 988-7455 Prerequistes: Placement test score on TSI or

More information

ACTL5103 Stochastic Modelling For Actuaries. Course Outline Semester 2, 2014

ACTL5103 Stochastic Modelling For Actuaries. Course Outline Semester 2, 2014 UNSW Australia Business School School of Risk and Actuarial Studies ACTL5103 Stochastic Modelling For Actuaries Course Outline Semester 2, 2014 Part A: Course-Specific Information Please consult Part B

More information

Bittinger, M. L., Ellenbogen, D. J., & Johnson, B. L. (2012). Prealgebra (6th ed.). Boston, MA: Addison-Wesley.

Bittinger, M. L., Ellenbogen, D. J., & Johnson, B. L. (2012). Prealgebra (6th ed.). Boston, MA: Addison-Wesley. Course Syllabus Course Description Explores the basic fundamentals of college-level mathematics. (Note: This course is for institutional credit only and will not be used in meeting degree requirements.

More information

Mathematics process categories

Mathematics process categories Mathematics process categories All of the UK curricula define multiple categories of mathematical proficiency that require students to be able to use and apply mathematics, beyond simple recall of facts

More information

Math 96: Intermediate Algebra in Context

Math 96: Intermediate Algebra in Context : Intermediate Algebra in Context Syllabus Spring Quarter 2016 Daily, 9:20 10:30am Instructor: Lauri Lindberg Office Hours@ tutoring: Tutoring Center (CAS-504) 8 9am & 1 2pm daily STEM (Math) Center (RAI-338)

More information

Course Name: Elementary Calculus Course Number: Math 2103 Semester: Fall Phone:

Course Name: Elementary Calculus Course Number: Math 2103 Semester: Fall Phone: Course Name: Elementary Calculus Course Number: Math 2103 Semester: Fall 2011 Instructor s Name: Ricky Streight Hours Credit: 3 Phone: 405-945-6794 email: ricky.streight@okstate.edu 1. COURSE: Math 2103

More information

AP Calculus AB. Nevada Academic Standards that are assessable at the local level only.

AP Calculus AB. Nevada Academic Standards that are assessable at the local level only. Calculus AB Priority Keys Aligned with Nevada Standards MA I MI L S MA represents a Major content area. Any concept labeled MA is something of central importance to the entire class/curriculum; it is a

More information

BENCHMARK MA.8.A.6.1. Reporting Category

BENCHMARK MA.8.A.6.1. Reporting Category Grade MA..A.. Reporting Category BENCHMARK MA..A.. Number and Operations Standard Supporting Idea Number and Operations Benchmark MA..A.. Use exponents and scientific notation to write large and small

More information

Math 121 Fundamentals of Mathematics I

Math 121 Fundamentals of Mathematics I I. Course Description: Math 121 Fundamentals of Mathematics I Math 121 is a general course in the fundamentals of mathematics. It includes a study of concepts of numbers and fundamental operations with

More information

Syllabus ENGR 190 Introductory Calculus (QR)

Syllabus ENGR 190 Introductory Calculus (QR) Syllabus ENGR 190 Introductory Calculus (QR) Catalog Data: ENGR 190 Introductory Calculus (4 credit hours). Note: This course may not be used for credit toward the J.B. Speed School of Engineering B. S.

More information

Communication and Cybernetics 17

Communication and Cybernetics 17 Communication and Cybernetics 17 Editors: K. S. Fu W. D. Keidel W. J. M. Levelt H. Wolter Communication and Cybernetics Editors: K.S.Fu, W.D.Keidel, W.1.M.Levelt, H.Wolter Vol. Vol. 2 Vol. 3 Vol. 4 Vol.

More information

CAAP. Content Analysis Report. Sample College. Institution Code: 9011 Institution Type: 4-Year Subgroup: none Test Date: Spring 2011

CAAP. Content Analysis Report. Sample College. Institution Code: 9011 Institution Type: 4-Year Subgroup: none Test Date: Spring 2011 CAAP Content Analysis Report Institution Code: 911 Institution Type: 4-Year Normative Group: 4-year Colleges Introduction This report provides information intended to help postsecondary institutions better

More information

EDEXCEL FUNCTIONAL SKILLS PILOT TEACHER S NOTES. Maths Level 2. Chapter 4. Working with measures

EDEXCEL FUNCTIONAL SKILLS PILOT TEACHER S NOTES. Maths Level 2. Chapter 4. Working with measures EDEXCEL FUNCTIONAL SKILLS PILOT TEACHER S NOTES Maths Level 2 Chapter 4 Working with measures SECTION G 1 Time 2 Temperature 3 Length 4 Weight 5 Capacity 6 Conversion between metric units 7 Conversion

More information

International Series in Operations Research & Management Science

International Series in Operations Research & Management Science International Series in Operations Research & Management Science Volume 240 Series Editor Camille C. Price Stephen F. Austin State University, TX, USA Associate Series Editor Joe Zhu Worcester Polytechnic

More information

MMOG Subscription Business Models: Table of Contents

MMOG Subscription Business Models: Table of Contents DFC Intelligence DFC Intelligence Phone 858-780-9680 9320 Carmel Mountain Rd Fax 858-780-9671 Suite C www.dfcint.com San Diego, CA 92129 MMOG Subscription Business Models: Table of Contents November 2007

More information

To link to this article: PLEASE SCROLL DOWN FOR ARTICLE

To link to this article:  PLEASE SCROLL DOWN FOR ARTICLE This article was downloaded by: [Dr Brian Winkel] On: 19 November 2014, At: 04:59 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer

More information

Using Calculators for Students in Grades 9-12: Geometry. Re-published with permission from American Institutes for Research

Using Calculators for Students in Grades 9-12: Geometry. Re-published with permission from American Institutes for Research Using Calculators for Students in Grades 9-12: Geometry Re-published with permission from American Institutes for Research Using Calculators for Students in Grades 9-12: Geometry By: Center for Implementing

More information

Grading Policy/Evaluation: The grades will be counted in the following way: Quizzes 30% Tests 40% Final Exam: 30%

Grading Policy/Evaluation: The grades will be counted in the following way: Quizzes 30% Tests 40% Final Exam: 30% COURSE SYLLABUS FALL 2010 MATH 0408 INTERMEDIATE ALGEBRA Course # 0408.06 Course Schedule/Location: TT 09:35 11:40, A-228 Instructor: Dr. Calin Agut, Office: J-202, Department of Mathematics, Brazosport

More information

Math 181, Calculus I

Math 181, Calculus I Math 181, Calculus I [Semester] [Class meeting days/times] [Location] INSTRUCTOR INFORMATION: Name: Office location: Office hours: Mailbox: Phone: Email: Required Material and Access: Textbook: Stewart,

More information

B.S/M.A in Mathematics

B.S/M.A in Mathematics B.S/M.A in Mathematics The dual Bachelor of Science/Master of Arts in Mathematics program provides an opportunity for individuals to pursue advanced study in mathematics and to develop skills that can

More information

SURVIVING ON MARS WITH GEOGEBRA

SURVIVING ON MARS WITH GEOGEBRA SURVIVING ON MARS WITH GEOGEBRA Lindsey States and Jenna Odom Miami University, OH Abstract: In this paper, the authors describe an interdisciplinary lesson focused on determining how long an astronaut

More information

MTH 141 Calculus 1 Syllabus Spring 2017

MTH 141 Calculus 1 Syllabus Spring 2017 Instructor: Section/Meets Office Hrs: Textbook: Calculus: Single Variable, by Hughes-Hallet et al, 6th ed., Wiley. Also needed: access code to WileyPlus (included in new books) Calculator: Not required,

More information

Focus of the Unit: Much of this unit focuses on extending previous skills of multiplication and division to multi-digit whole numbers.

Focus of the Unit: Much of this unit focuses on extending previous skills of multiplication and division to multi-digit whole numbers. Approximate Time Frame: 3-4 weeks Connections to Previous Learning: In fourth grade, students fluently multiply (4-digit by 1-digit, 2-digit by 2-digit) and divide (4-digit by 1-digit) using strategies

More information

Math-U-See Correlation with the Common Core State Standards for Mathematical Content for Third Grade

Math-U-See Correlation with the Common Core State Standards for Mathematical Content for Third Grade Math-U-See Correlation with the Common Core State Standards for Mathematical Content for Third Grade The third grade standards primarily address multiplication and division, which are covered in Math-U-See

More information

SOUTHERN MAINE COMMUNITY COLLEGE South Portland, Maine 04106

SOUTHERN MAINE COMMUNITY COLLEGE South Portland, Maine 04106 SOUTHERN MAINE COMMUNITY COLLEGE South Portland, Maine 04106 Title: Precalculus Catalog Number: MATH 190 Credit Hours: 3 Total Contact Hours: 45 Instructor: Gwendolyn Blake Email: gblake@smccme.edu Website:

More information

Extending Place Value with Whole Numbers to 1,000,000

Extending Place Value with Whole Numbers to 1,000,000 Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Recognize that a digit

More information

Detailed course syllabus

Detailed course syllabus Detailed course syllabus 1. Linear regression model. Ordinary least squares method. This introductory class covers basic definitions of econometrics, econometric model, and economic data. Classification

More information

TOPICS LEARNING OUTCOMES ACTIVITES ASSESSMENT Numbers and the number system

TOPICS LEARNING OUTCOMES ACTIVITES ASSESSMENT Numbers and the number system Curriculum Overview Mathematics 1 st term 5º grade - 2010 TOPICS LEARNING OUTCOMES ACTIVITES ASSESSMENT Numbers and the number system Multiplies and divides decimals by 10 or 100. Multiplies and divide

More information

GCSE. Mathematics A. Mark Scheme for January General Certificate of Secondary Education Unit A503/01: Mathematics C (Foundation Tier)

GCSE. Mathematics A. Mark Scheme for January General Certificate of Secondary Education Unit A503/01: Mathematics C (Foundation Tier) GCSE Mathematics A General Certificate of Secondary Education Unit A503/0: Mathematics C (Foundation Tier) Mark Scheme for January 203 Oxford Cambridge and RSA Examinations OCR (Oxford Cambridge and RSA)

More information

MARE Publication Series

MARE Publication Series MARE Publication Series Volume 8 Series Editors Maarten Bavinck University of Amsterdam, Amsterdam, The Netherlands Svein Jentoft Tromsø, Norway The MARE Publication Series is an initiative of the Centre

More information

Are You Ready? Simplify Fractions

Are You Ready? Simplify Fractions SKILL 10 Simplify Fractions Teaching Skill 10 Objective Write a fraction in simplest form. Review the definition of simplest form with students. Ask: Is 3 written in simplest form? Why 7 or why not? (Yes,

More information

EDEXCEL FUNCTIONAL SKILLS PILOT. Maths Level 2. Chapter 7. Working with probability

EDEXCEL FUNCTIONAL SKILLS PILOT. Maths Level 2. Chapter 7. Working with probability Working with probability 7 EDEXCEL FUNCTIONAL SKILLS PILOT Maths Level 2 Chapter 7 Working with probability SECTION K 1 Measuring probability 109 2 Experimental probability 111 3 Using tables to find the

More information

Probability and Statistics Curriculum Pacing Guide

Probability and Statistics Curriculum Pacing Guide Unit 1 Terms PS.SPMJ.3 PS.SPMJ.5 Plan and conduct a survey to answer a statistical question. Recognize how the plan addresses sampling technique, randomization, measurement of experimental error and methods

More information

Big Ideas Math Grade 6 Answer Key

Big Ideas Math Grade 6 Answer Key Big Ideas Math Grade 6 Answer Key Free PDF ebook Download: Big Ideas Math Grade 6 Answer Key Download or Read Online ebook big ideas math grade 6 answer key in PDF Format From The Best User Guide Database

More information

Measurement & Analysis in the Real World

Measurement & Analysis in the Real World Measurement & Analysis in the Real World Tools for Cleaning Messy Data Will Hayes SEI Robert Stoddard SEI Rhonda Brown SEI Software Solutions Conference 2015 November 16 18, 2015 Copyright 2015 Carnegie

More information

PRODUCT PLATFORM AND PRODUCT FAMILY DESIGN

PRODUCT PLATFORM AND PRODUCT FAMILY DESIGN PRODUCT PLATFORM AND PRODUCT FAMILY DESIGN PRODUCT PLATFORM AND PRODUCT FAMILY DESIGN Methods and Applications Edited by Timothy W. Simpson 1, Zahed Siddique 2, and Jianxin (Roger) Jiao 3 1 The Pennsylvania

More information

South Carolina English Language Arts

South Carolina English Language Arts South Carolina English Language Arts A S O F J U N E 2 0, 2 0 1 0, T H I S S TAT E H A D A D O P T E D T H E CO M M O N CO R E S TAT E S TA N DA R D S. DOCUMENTS REVIEWED South Carolina Academic Content

More information

School of Innovative Technologies and Engineering

School of Innovative Technologies and Engineering School of Innovative Technologies and Engineering Department of Applied Mathematical Sciences Proficiency Course in MATLAB COURSE DOCUMENT VERSION 1.0 PCMv1.0 July 2012 University of Technology, Mauritius

More information

LLD MATH. Student Eligibility: Grades 6-8. Credit Value: Date Approved: 8/24/15

LLD MATH. Student Eligibility: Grades 6-8. Credit Value: Date Approved: 8/24/15 PUBLIC SCHOOLS OF EDISON TOWNSHIP DIVISION OF CURRICULUM AND INSTRUCTION LLD MATH Length of Course: Elective/Required: School: Full Year Required Middle Schools Student Eligibility: Grades 6-8 Credit Value:

More information

Excel Formulas & Functions

Excel Formulas & Functions Microsoft Excel Formulas & Functions 4th Edition Microsoft Excel Formulas & Functions 4th Edition by Ken Bluttman Microsoft Excel Formulas & Functions For Dummies, 4th Edition Published by: John Wiley

More information

Course Goal This is the final course in the developmental mathematics sequence and its purpose is to prepare students for College Algebra.

Course Goal This is the final course in the developmental mathematics sequence and its purpose is to prepare students for College Algebra. Mathematics ONLINE Math 0312: Intermediate Algebra CRN 86448 Spring 2 nd Eight Weeks/2016 3 hour lecture course +1hour lab / 64 hours per semester/ 8 weeks Textbook: Introductory and Intermediate Algebra.

More information

Perspectives of Information Systems

Perspectives of Information Systems Perspectives of Information Systems Springer-Science+ Business Media, LLC Vesa Savolainen Editor and Main Author Perspectives of Information Systems Springer Vesa Savolainen Department of Computer Science

More information

Let s think about how to multiply and divide fractions by fractions!

Let s think about how to multiply and divide fractions by fractions! Let s think about how to multiply and divide fractions by fractions! June 25, 2007 (Monday) Takehaya Attached Elementary School, Tokyo Gakugei University Grade 6, Class # 1 (21 boys, 20 girls) Instructor:

More information

Pearson Grade 8 Practice And Homework

Pearson Grade 8 Practice And Homework Pearson Grade 8 And Free PDF ebook Download: Pearson Grade 8 And Download or Read Online ebook pearson grade 8 practice and homework in PDF Format From The Best User Guide Database 3rd Grade EOG Prep.

More information

Page 1 of 11. Curriculum Map: Grade 4 Math Course: Math 4 Sub-topic: General. Grade(s): None specified

Page 1 of 11. Curriculum Map: Grade 4 Math Course: Math 4 Sub-topic: General. Grade(s): None specified Curriculum Map: Grade 4 Math Course: Math 4 Sub-topic: General Grade(s): None specified Unit: Creating a Community of Mathematical Thinkers Timeline: Week 1 The purpose of the Establishing a Community

More information

Dublin City Schools Mathematics Graded Course of Study GRADE 4

Dublin City Schools Mathematics Graded Course of Study GRADE 4 I. Content Standard: Number, Number Sense and Operations Standard Students demonstrate number sense, including an understanding of number systems and reasonable estimates using paper and pencil, technology-supported

More information

Medical Complexity: A Pragmatic Theory

Medical Complexity: A Pragmatic Theory http://eoimages.gsfc.nasa.gov/images/imagerecords/57000/57747/cloud_combined_2048.jpg Medical Complexity: A Pragmatic Theory Chris Feudtner, MD PhD MPH The Children s Hospital of Philadelphia Main Thesis

More information

Cheat The Jlab Algebra 2

Cheat The Jlab Algebra 2 Cheat The Jlab Algebra 2 Free PDF ebook Download: Cheat The Jlab Algebra 2 Download or Read Online ebook cheat the jlab algebra 2 in PDF Format From The Best User Guide Database Jun 12, 2011 - Router Accounts.

More information

Grade 5 + DIGITAL. EL Strategies. DOK 1-4 RTI Tiers 1-3. Flexible Supplemental K-8 ELA & Math Online & Print

Grade 5 + DIGITAL. EL Strategies. DOK 1-4 RTI Tiers 1-3. Flexible Supplemental K-8 ELA & Math Online & Print Standards PLUS Flexible Supplemental K-8 ELA & Math Online & Print Grade 5 SAMPLER Mathematics EL Strategies DOK 1-4 RTI Tiers 1-3 15-20 Minute Lessons Assessments Consistent with CA Testing Technology

More information

Economics 201 Principles of Microeconomics Fall 2010 MWF 10:00 10:50am 160 Bryan Building

Economics 201 Principles of Microeconomics Fall 2010 MWF 10:00 10:50am 160 Bryan Building Economics 201 Principles of Microeconomics Fall 2010 MWF 10:00 10:50am 160 Bryan Building Professor: Dr. Michelle Sheran Office: 445 Bryan Building Phone: 256-1192 E-mail: mesheran@uncg.edu Office Hours:

More information

TABE 9&10. Revised 8/2013- with reference to College and Career Readiness Standards

TABE 9&10. Revised 8/2013- with reference to College and Career Readiness Standards TABE 9&10 Revised 8/2013- with reference to College and Career Readiness Standards LEVEL E Test 1: Reading Name Class E01- INTERPRET GRAPHIC INFORMATION Signs Maps Graphs Consumer Materials Forms Dictionary

More information

Mathematics Success Grade 7

Mathematics Success Grade 7 T894 Mathematics Success Grade 7 [OBJECTIVE] The student will find probabilities of compound events using organized lists, tables, tree diagrams, and simulations. [PREREQUISITE SKILLS] Simple probability,

More information

CS/SE 3341 Spring 2012

CS/SE 3341 Spring 2012 CS/SE 3341 Spring 2012 Probability and Statistics in Computer Science & Software Engineering (Section 001) Instructor: Dr. Pankaj Choudhary Meetings: TuTh 11 30-12 45 p.m. in ECSS 2.412 Office: FO 2.408-B

More information

UNIT ONE Tools of Algebra

UNIT ONE Tools of Algebra UNIT ONE Tools of Algebra Subject: Algebra 1 Grade: 9 th 10 th Standards and Benchmarks: 1 a, b,e; 3 a, b; 4 a, b; Overview My Lessons are following the first unit from Prentice Hall Algebra 1 1. Students

More information

Exploring Derivative Functions using HP Prime

Exploring Derivative Functions using HP Prime Exploring Derivative Functions using HP Prime Betty Voon Wan Niu betty@uniten.edu.my College of Engineering Universiti Tenaga Nasional Malaysia Wong Ling Shing Faculty of Health and Life Sciences, INTI

More information

Common Core Standards Alignment Chart Grade 5

Common Core Standards Alignment Chart Grade 5 Common Core Standards Alignment Chart Grade 5 Units 5.OA.1 5.OA.2 5.OA.3 5.NBT.1 5.NBT.2 5.NBT.3 5.NBT.4 5.NBT.5 5.NBT.6 5.NBT.7 5.NF.1 5.NF.2 5.NF.3 5.NF.4 5.NF.5 5.NF.6 5.NF.7 5.MD.1 5.MD.2 5.MD.3 5.MD.4

More information

1 Use complex features of a word processing application to a given brief. 2 Create a complex document. 3 Collaborate on a complex document.

1 Use complex features of a word processing application to a given brief. 2 Create a complex document. 3 Collaborate on a complex document. National Unit specification General information Unit code: HA6M 46 Superclass: CD Publication date: May 2016 Source: Scottish Qualifications Authority Version: 02 Unit purpose This Unit is designed to

More information

Cambridge NATIONALS. Creative imedia Level 1/2. UNIT R081 - Pre-Production Skills DELIVERY GUIDE

Cambridge NATIONALS. Creative imedia Level 1/2. UNIT R081 - Pre-Production Skills DELIVERY GUIDE Cambridge NATIONALS Creative imedia Level 1/2 UNIT R081 - Pre-Production Skills VERSION 1 APRIL 2013 INDEX Introduction Page 3 Unit R081 - Pre-Production Skills Page 4 Learning Outcome 1 - Understand the

More information

Written by Wendy Osterman

Written by Wendy Osterman Pre-Algebra Written by Wendy Osterman Editor: Alaska Hults Illustrator: Corbin Hillam Designer/Production: Moonhee Pak/Cari Helstrom Cover Designer: Barbara Peterson Art Director: Tom Cochrane Project

More information

Functional Skills Mathematics Level 2 assessment

Functional Skills Mathematics Level 2 assessment Functional Skills Mathematics Level 2 assessment www.cityandguilds.com September 2015 Version 1.0 Marking scheme ONLINE V2 Level 2 Sample Paper 4 Mark Represent Analyse Interpret Open Fixed S1Q1 3 3 0

More information

Availability of Grants Largely Offset Tuition Increases for Low-Income Students, U.S. Report Says

Availability of Grants Largely Offset Tuition Increases for Low-Income Students, U.S. Report Says Wednesday, October 2, 2002 http://chronicle.com/daily/2002/10/2002100206n.htm Availability of Grants Largely Offset Tuition Increases for Low-Income Students, U.S. Report Says As the average price of attending

More information

Pre-Algebra A. Syllabus. Course Overview. Course Goals. General Skills. Credit Value

Pre-Algebra A. Syllabus. Course Overview. Course Goals. General Skills. Credit Value Syllabus Pre-Algebra A Course Overview Pre-Algebra is a course designed to prepare you for future work in algebra. In Pre-Algebra, you will strengthen your knowledge of numbers as you look to transition

More information

Problem Solving for Success Handbook. Solve the Problem Sustain the Solution Celebrate Success

Problem Solving for Success Handbook. Solve the Problem Sustain the Solution Celebrate Success Problem Solving for Success Handbook Solve the Problem Sustain the Solution Celebrate Success Problem Solving for Success Handbook Solve the Problem Sustain the Solution Celebrate Success Rod Baxter 2015

More information

FractionWorks Correlation to Georgia Performance Standards

FractionWorks Correlation to Georgia Performance Standards Cheryl Keck Educational Sales Consultant Phone: 800-445-5985 ext. 3231 ckeck@etacuisenaire.com www.etacuisenaire.com FractionWorks Correlation to Georgia Performance s Correlated to Georgia Performance

More information

Mathematics SPA Report Section I Context

Mathematics SPA Report Section I Context MathematicsSPAReport SectionI Context StateandInstitutionalPolicies Threestateagencieslargelyoverseethepreparationandcertificationofnewteachers.TheTexasEducationAgency(TEA)isresponsiblefor providingleadership,guidance,andresourcestohelpschoolsmeettheeducationalneedsofallstudents,andistheprimaryadministrativeunit

More information

Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking

Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking Catherine Pearn The University of Melbourne Max Stephens The University of Melbourne

More information

Instrumentation, Control & Automation Staffing. Maintenance Benchmarking Study

Instrumentation, Control & Automation Staffing. Maintenance Benchmarking Study Electronic Document Instrumentation, Control & Automation Staffing Prepared by ITA Technical Committee, Maintenance Subcommittee, Task Force on IC&A Staffing John Petito, Chair Richard Haugh, Vice-Chair

More information

EGRHS Course Fair. Science & Math AP & IB Courses

EGRHS Course Fair. Science & Math AP & IB Courses EGRHS Course Fair Science & Math AP & IB Courses Science Courses: AP Physics IB Physics SL IB Physics HL AP Biology IB Biology HL AP Physics Course Description Course Description AP Physics C (Mechanics)

More information

Paper Reference. Edexcel GCSE Mathematics (Linear) 1380 Paper 1 (Non-Calculator) Foundation Tier. Monday 6 June 2011 Afternoon Time: 1 hour 30 minutes

Paper Reference. Edexcel GCSE Mathematics (Linear) 1380 Paper 1 (Non-Calculator) Foundation Tier. Monday 6 June 2011 Afternoon Time: 1 hour 30 minutes Centre No. Candidate No. Paper Reference 1 3 8 0 1 F Paper Reference(s) 1380/1F Edexcel GCSE Mathematics (Linear) 1380 Paper 1 (Non-Calculator) Foundation Tier Monday 6 June 2011 Afternoon Time: 1 hour

More information

Diagnostic Test. Middle School Mathematics

Diagnostic Test. Middle School Mathematics Diagnostic Test Middle School Mathematics Copyright 2010 XAMonline, Inc. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by

More information

THE UNITED REPUBLIC OF TANZANIA MINISTRY OF EDUCATION, SCIENCE, TECHNOLOGY AND VOCATIONAL TRAINING CURRICULUM FOR BASIC EDUCATION STANDARD I AND II

THE UNITED REPUBLIC OF TANZANIA MINISTRY OF EDUCATION, SCIENCE, TECHNOLOGY AND VOCATIONAL TRAINING CURRICULUM FOR BASIC EDUCATION STANDARD I AND II THE UNITED REPUBLIC OF TANZANIA MINISTRY OF EDUCATION, SCIENCE, TECHNOLOGY AND VOCATIONAL TRAINING CURRICULUM FOR BASIC EDUCATION STANDARD I AND II 2016 Ministry of Education, Science,Technology and Vocational

More information

S T A T 251 C o u r s e S y l l a b u s I n t r o d u c t i o n t o p r o b a b i l i t y

S T A T 251 C o u r s e S y l l a b u s I n t r o d u c t i o n t o p r o b a b i l i t y Department of Mathematics, Statistics and Science College of Arts and Sciences Qatar University S T A T 251 C o u r s e S y l l a b u s I n t r o d u c t i o n t o p r o b a b i l i t y A m e e n A l a

More information

A Practical Introduction to Teacher Training in ELT

A Practical Introduction to Teacher Training in ELT Teaching English A Practical Introduction to Teacher Training in ELT John Hughes Packed with practical advice, training tips, and workshop ideas A Practical Introduction to Teacher Training in ELT John

More information