# American Psychologist

Size: px
Start display at page:

## Transcription

1 American Psychologist The Complex Dynamics of Wishful Thinking: The Critical Positivity Ratio Nicholas J. L. Brown, Alan D. Sokal, and Harris L. Friedman Online First Publication, July 15, doi: /a CITATION Brown, N. J. L., Sokal, A. D., & Friedman, H. L. (2013, July 15). The Complex Dynamics of Wishful Thinking: The Critical Positivity Ratio. American Psychologist. Advance online publication. doi: /a

4 is always determined, at least in principle, from the initial conditions, even though it may not be given by any simple explicit formula. Example 2: Consider a population x of some species living in a limited territory, with a maximum sustainable population X max. Then a plausible (though of course extremely oversimplified) model for the growth of this population is given by the differential equation Alan D. Sokal additional variables), and only on the value of x at that same moment of time (i.e., not on the values in the past). (DE5) The rate of change of x at time t is exactly F x t, where F is an explicitly specified function. The mathematical model of the phenomenon under study thus consists of Equation 1 together with the specification of the function F. Example 1: Consider a bank account with continuously compounded interest. Then the amount of money in the account (x) increases with time (t) according to the differential equation dx rx (2) dt where r is the interest rate. This has the form of Equation 1 with F x rx. This same equation describes the cooling of a coffee cup; the decay of radioactive atoms; and a vast number of other physical, biological, and social phenomena. (In some of these applications, r is a negative number.) Equation 2 is an example of a linear differential equation, because the function F x rx is linear (i.e., doubling x causes F x to precisely double). Equation 2 happens to have a simple solution, namely x t x 0 e rt, where x 0 is the account balance at time 0, and e is the base of natural logarithms. This formula illustrates an important general principle: The solution of a differential equation is completely determined by the initial conditions, that is, the value(s) of the dependent variable(s) at time 0. It should be stressed that most differential equations do not have simple solutions that can be explicitly written down; rather, the solutions have to be studied numerically, by computer. Nevertheless, the solution at arbitrary time t dx dt rx 1 x X max (3) where r is some positive number, which has the form of Equation 1 with F x rx 1 x X max. Equation 3 is an example of a nonlinear differential equation, because here the function F is nonlinear (i.e., doubling x does not cause F x to precisely double). Of course, population is not, strictly speaking, a continuous variable, since there is no such thing as a fractional person. This would appear to conflict with principle DE1 above. However, if the population x is large (e.g., millions of people), then only a negligible error is made by treating the population as if it were a continuous variable. Several dependent variables. Let us now consider the case in which we have several dependent variables. We assume once again that both the independent variable t and the dependent variables x 1,...,x n can be treated as continuous quantities, and that x 1,...,x n vary smoothly as a function of t. Then a system of (first-order) differential equations for the functions x 1 t,...,x n t is a system of equations of the form dx 1 dt F 1 x 1,..., x n É dx n dt F n x 1,..., x n where F 1,...,F n are specified functions. The model is constituted by the system of Equation 4 together with the specification of the functions F 1,...,F n. Example 3: Lorenz (1963), building on work by Saltzman (1962), introduced the following system of differential equations as a simplified model of convective flow in fluids: dx X Y d (5a) dy rx Y XZ d (5b) dz bz XY d (5c) Here, the independent variable is a dimensionless variable that is proportional to time t. The dependent variables (4) Month 2013 American Psychologist 3

5 Harris L. Friedman X, Y, and Z are also dimensionless and represent various aspects of the fluid s motion and its temperature gradients. (In the solutions studied by Lorenz, X and Y oscillate between positive and negative values, while Z stays positive.) Finally,, b, and r are positive dimensionless parameters that characterize certain properties of the fluid and its flow; in particular, r measures (roughly speaking) the strength of the tendency to develop convection. For future reference, let us explain what is meant by dimensionless. A quantity is called dimensionful if its numerical value depends on an arbitrary choice of units. For instance, lengths (measured in meters or furlongs or...) are dimensionful, as are times and masses. By contrast, a quantity is called dimensionless if its numerical value does not depend on a choice of units. For instance, the ratio of two lengths is dimensionless, because the units cancel out when forming the ratio; likewise for the ratio of two times or of two masses. Physical laws are best expressed in terms of dimensionless quantities, since these are independent of the choice of units; this explains why Lorenz (1963) rescaled his equations as he did. In particular, Lorenz s variable equals t T, where T is a particular characteristic time in the fluid-flow problem. Nonlinear Dynamics and Chaos The mathematical field of nonlinear dynamics popularly known as chaos theory is founded on the observation that simple equations can under some circumstances have extremely complicated solutions. In particular, fairly simple systems of nonlinear differential equations can exhibit sensitive dependence to initial conditions; that is, small changes in the initial conditions can lead to deviations in the subsequent trajectory that grow exponentially over time. Such systems, while deterministic in principle, can be unpredictable in practice beyond a limited window of time; the behavior can appear random even though it is not. We refer the reader to Lorenz (1993), Stewart (1997), and Williams (1997) for sober nontechnical introductions to nonlinear dynamics, and to Strogatz (1994) and Hilborn (2000) for introductions presuming some background in undergraduate mathematics and physics. The Lorenz system, revisited. The Lorenz system (Equations 5a 5c above) illustrates nicely some of the concepts of nonlinear dynamics. First, this system has a fixed point at X Y Z 0: If the system is started there, it stays there forever. (Physically, this fixed point corresponds to a fluid at rest.) For r 1 it turns out that this fixed point is stable: If the system is started near the fixed point, it will move toward the fixed point as time goes on. For r 1 this fixed point is unstable: If started near the fixed point, the system will move away from it. For r 1 there is another pair of fixed points, at X Y b r 1, Z r 1. (Physically, these fixed points correspond to a steady-state convective flow.) It turns out that these fixed points are stable for r r crit and are unstable for r r crit, where r crit b 3 b 1 and we assume b 1. What happens for r r crit? Lorenz (1963) investigated the trajectories numerically and found that they tend to a butterfly-shaped set now known as the Lorenz attractor. This set is a fractal: It is neither two-dimensional (a surface) nor three-dimensional (a volume) but something inbetween. Moreover, the trajectories near the attractor exhibit sensitive dependence to initial conditions (i.e., they are chaotic). 2 One final remark: Mathematicians studying the Lorenz system typically (though not always) fix and b and vary r. Saltzman (1962) chose the illustrative values 10 and b 8 3; then Lorenz (1963) followed him, as have most (though not all) workers ever since. However, there is nothing magical about these values; indeed, any other values within a fairly wide range would produce qualitatively similar behavior (Sparrow, 1982, pp ). When Can Differential Equations Validly Be Applied? When used properly, differential equations constitute a powerful tool for modeling time-dependent phenomena in the natural and social sciences. However, a number of preconditions must be met if such an application is to be valid. In order to apply differential equations to a specific natural or social system, one must first do the following: (VA1) Identify and define precisely the variables that specify the state of the system at a given moment of time. Per principles DE1 and DE2 (above), these variables must be continuous (or at least approximately so), not discrete, 2 Actually, the Lorenz attractor is born at a value r A slightly less than r crit ; for r A r r crit, both the fixed points and the Lorenz attractor are stable (Sparrow, 1982, pp ). 4 Month 2013 American Psychologist

### The Complex Dynamics of Wishful Thinking: The Critical Positivity Ratio

The Complex Dynamics of Wishful Thinking: The Critical Positivity Ratio Nicholas J. L. Brown Strasbourg, France Alan D. Sokal New York University and University College London Harris L. Friedman Saybrook

### Positive Psychology and Romantic Scientism: Reply to Comments on Brown, Sokal, & Friedman (2013)

Positive Psychology and Romantic Scientism: Reply to Comments on Brown, Sokal, & Friedman (2013) Nicholas J. L. Brown New School of Psychotherapy and Counselling Alan D. Sokal New York University and University

### PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*

Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### CHAOS LIMITATION OR EVEN END OF SUPPLY CHAIN MANAGEMENT

CHAOS LIMITATION OR EVEN END OF SUPPLY CHAIN MANAGEMENT Michael Grabinski 1 Abstract Proven in the early 196s, weather forecast is not possible for an arbitrarily long period of time for principle reasons.

### For example, estimate the population of the United States as 3 times 10⁸ and the

CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### Performance Level Descriptors Grade 6 Mathematics

Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.1-2 Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with

### Entry Level College Mathematics: Algebra or Modeling

Entry Level College Mathematics: Algebra or Modeling Dan Kalman Dan Kalman is Associate Professor in Mathematics and Statistics at American University. His interests include matrix theory, curriculum development,

### What Is School Mathematics?

What Is School Mathematics? Lisbon, Portugal January 30, 2010 H. Wu *I am grateful to Alexandra Alves-Rodrigues for her many contributions that helped shape this document. The German conductor Herbert

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

### THE STATISTICAL TREATMENT OF EXPERIMENTAL DATA 1

THE STATISTICAL TREATMET OF EXPERIMETAL DATA Introduction The subject of statistical data analysis is regarded as crucial by most scientists, since error-free measurement is impossible in virtually all

### Writing the Empirical Social Science Research Paper: A Guide for the Perplexed. Josh Pasek. University of Michigan.

Writing the Empirical Social Science Research Paper: A Guide for the Perplexed Josh Pasek University of Michigan January 24, 2012 Correspondence about this manuscript should be addressed to Josh Pasek,

### Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis

Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................

### Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### Analyzing Research Articles: A Guide for Readers and Writers 1. Sam Mathews, Ph.D. Department of Psychology The University of West Florida

Analyzing Research Articles: A Guide for Readers and Writers 1 Sam Mathews, Ph.D. Department of Psychology The University of West Florida The critical reader of a research report expects the writer to

### The Basics of Interest Theory

Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

### CFSD 21 ST CENTURY SKILL RUBRIC CRITICAL & CREATIVE THINKING

Critical and creative thinking (higher order thinking) refer to a set of cognitive skills or strategies that increases the probability of a desired outcome. In an information- rich society, the quality

### Problem of the Month: Perfect Pair

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### Methodological Issues for Interdisciplinary Research

J. T. M. Miller, Department of Philosophy, University of Durham 1 Methodological Issues for Interdisciplinary Research Much of the apparent difficulty of interdisciplinary research stems from the nature

### 2.2. Instantaneous Velocity

2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### Inequality, Mobility and Income Distribution Comparisons

Fiscal Studies (1997) vol. 18, no. 3, pp. 93 30 Inequality, Mobility and Income Distribution Comparisons JOHN CREEDY * Abstract his paper examines the relationship between the cross-sectional and lifetime

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### Problem Solving and Data Analysis

Chapter 20 Problem Solving and Data Analysis The Problem Solving and Data Analysis section of the SAT Math Test assesses your ability to use your math understanding and skills to solve problems set in

### Berkeley CS191x: Quantum Mechanics and Quantum Computation Optional Class Project

Berkeley CS191x: Quantum Mechanics and Quantum Computation Optional Class Project This document describes the optional class project for the Fall 2013 offering of CS191x. The project will not be graded.

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

### Computational Finance Options

1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to

### Using simulation to calculate the NPV of a project

Using simulation to calculate the NPV of a project Marius Holtan Onward Inc. 5/31/2002 Monte Carlo simulation is fast becoming the technology of choice for evaluating and analyzing assets, be it pure financial

### The Alignment of Common Core and ACT s College and Career Readiness System. June 2010

The Alignment of Common Core and ACT s College and Career Readiness System June 2010 ACT is an independent, not-for-profit organization that provides assessment, research, information, and program management

### 15.062 Data Mining: Algorithms and Applications Matrix Math Review

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### DRAFT. Algebra 1 EOC Item Specifications

DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Experiment #1, Analyze Data using Excel, Calculator and Graphs.

Physics 182 - Fall 2014 - Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring

### What is the Purpose of College Algebra? Sheldon P. Gordon Farmingdale State College of New York

What is the Purpose of College Algebra? Sheldon P. Gordon Farmingdale State College of New York Each year, over 1,000,000 students [8] take College Algebra and related courses. Most of these courses were

### Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

### Standards for Mathematical Practice: Commentary and Elaborations for 6 8

Standards for Mathematical Practice: Commentary and Elaborations for 6 8 c Illustrative Mathematics 6 May 2014 Suggested citation: Illustrative Mathematics. (2014, May 6). Standards for Mathematical Practice:

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### The Mathematics School Teachers Should Know

The Mathematics School Teachers Should Know Lisbon, Portugal January 29, 2010 H. Wu *I am grateful to Alexandra Alves-Rodrigues for her many contributions that helped shape this document. Do school math

### Mathematics Curriculum Guide Precalculus 2015-16. Page 1 of 12

Mathematics Curriculum Guide Precalculus 2015-16 Page 1 of 12 Paramount Unified School District High School Math Curriculum Guides 2015 16 In 2015 16, PUSD will continue to implement the Standards by providing

### Appendix A: Science Practices for AP Physics 1 and 2

Appendix A: Science Practices for AP Physics 1 and 2 Science Practice 1: The student can use representations and models to communicate scientific phenomena and solve scientific problems. The real world

### 3 More on Accumulation and Discount Functions

3 More on Accumulation and Discount Functions 3.1 Introduction In previous section, we used 1.03) # of years as the accumulation factor. This section looks at other accumulation factors, including various

### Writing a Formal Lab Report

Writing a Formal Lab Report Note: This handout provides guidelines for writing a formal, typed laboratory report for a Biology, Chemistry, Natural Science, or Physics class. Routine lab write-ups such

### Introduction to time series analysis

Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Lecture 32: Systems of equations

Lecture 32: Systems of equations Nathan Pflueger 23 November 2011 1 Introduction So far in the differential equations unit, we have considered individual differential equations which describe how a single

### CHANCE ENCOUNTERS. Making Sense of Hypothesis Tests. Howard Fincher. Learning Development Tutor. Upgrade Study Advice Service

CHANCE ENCOUNTERS Making Sense of Hypothesis Tests Howard Fincher Learning Development Tutor Upgrade Study Advice Service Oxford Brookes University Howard Fincher 2008 PREFACE This guide has a restricted

### Evaluation Tool for Assessment Instrument Quality

REPRODUCIBLE Figure 4.4: Evaluation Tool for Assessment Instrument Quality Assessment indicators Description of Level 1 of the Indicator Are Not Present Limited of This Indicator Are Present Substantially

### Depth-of-Knowledge Levels for Four Content Areas Norman L. Webb March 28, 2002. Reading (based on Wixson, 1999)

Depth-of-Knowledge Levels for Four Content Areas Norman L. Webb March 28, 2002 Language Arts Levels of Depth of Knowledge Interpreting and assigning depth-of-knowledge levels to both objectives within

### How to Write a Formal Lab Report

Union College Physics and Astronomy How to Write a Formal Lab Report A formal lab report is essentially a scaled-down version of a scientific paper, reporting on the results of an experiment that you and

### Chapter 6: The Information Function 129. CHAPTER 7 Test Calibration

Chapter 6: The Information Function 129 CHAPTER 7 Test Calibration 130 Chapter 7: Test Calibration CHAPTER 7 Test Calibration For didactic purposes, all of the preceding chapters have assumed that the

### MATHEMATICS OF FINANCE AND INVESTMENT

MATHEMATICS OF FINANCE AND INVESTMENT G. I. FALIN Department of Probability Theory Faculty of Mechanics & Mathematics Moscow State Lomonosov University Moscow 119992 g.falin@mail.ru 2 G.I.Falin. Mathematics

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Basic Concepts in Research and Data Analysis

Basic Concepts in Research and Data Analysis Introduction: A Common Language for Researchers...2 Steps to Follow When Conducting Research...3 The Research Question... 3 The Hypothesis... 4 Defining the

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### The Mathematics of Alcoholics Anonymous

The Mathematics of Alcoholics Anonymous "As a celebrated American statesman put it, 'Let's look at the record. Bill Wilson, Alcoholics Anonymous, page 50, A.A.W.S. Inc., 2001. Part 2: A.A. membership surveys

### NODAL ANALYSIS. Circuits Nodal Analysis 1 M H Miller

NODAL ANALYSIS A branch of an electric circuit is a connection between two points in the circuit. In general a simple wire connection, i.e., a 'short-circuit', is not considered a branch since it is known

### Problem Solving Basics and Computer Programming

Problem Solving Basics and Computer Programming A programming language independent companion to Roberge/Bauer/Smith, "Engaged Learning for Programming in C++: A Laboratory Course", Jones and Bartlett Publishers,

### NEW MEXICO Grade 6 MATHEMATICS STANDARDS

PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

### You know from calculus that functions play a fundamental role in mathematics.

CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

### Week 7 - Game Theory and Industrial Organisation

Week 7 - Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number

### Interpretation of Test Scores for the ACCUPLACER Tests

Interpretation of Test Scores for the ACCUPLACER Tests ACCUPLACER is a trademark owned by the College Entrance Examination Board. Visit The College Board on the Web at: www.collegeboard.com/accuplacer

### Prentice Hall Connected Mathematics 2, 7th Grade Units 2009

Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Grade 7 C O R R E L A T E D T O from March 2009 Grade 7 Problem Solving Build new mathematical knowledge through problem solving. Solve problems

### Unit 1: Place value and operations with whole numbers and decimals

Unit 1: Place value and operations with whole numbers and decimals Content Area: Mathematics Course(s): Generic Course Time Period: 1st Marking Period Length: 10 Weeks Status: Published Unit Overview Students

### ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.

### Chapter 2 Conceptualizing Scientific Inquiry

Chapter 2 Conceptualizing Scientific Inquiry 2.1 Introduction In order to develop a strategy for the assessment of scientific inquiry in a laboratory setting, a theoretical construct of the components

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### PURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be

Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our

### Industry Environment and Concepts for Forecasting 1

Table of Contents Industry Environment and Concepts for Forecasting 1 Forecasting Methods Overview...2 Multilevel Forecasting...3 Demand Forecasting...4 Integrating Information...5 Simplifying the Forecast...6

### Global Seasonal Phase Lag between Solar Heating and Surface Temperature

Global Seasonal Phase Lag between Solar Heating and Surface Temperature Summer REU Program Professor Tom Witten By Abstract There is a seasonal phase lag between solar heating from the sun and the surface

### On Correlating Performance Metrics

On Correlating Performance Metrics Yiping Ding and Chris Thornley BMC Software, Inc. Kenneth Newman BMC Software, Inc. University of Massachusetts, Boston Performance metrics and their measurements are

### OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

### Levels of measurement in psychological research:

Research Skills: Levels of Measurement. Graham Hole, February 2011 Page 1 Levels of measurement in psychological research: Psychology is a science. As such it generally involves objective measurement of

### Format for Experiment Preparation and Write-Up

Format for Experiment Preparation and Write-Up Scientists try to answer questions by applying consistent, logical reasoning to describe, explain, and predict observations; and by performing experiments

### 0.75 75% ! 3 40% 0.65 65% Percent Cards. This problem gives you the chance to: relate fractions, decimals and percents

Percent Cards This problem gives you the chance to: relate fractions, decimals and percents Mrs. Lopez makes sets of cards for her math class. All the cards in a set have the same value. Set A 3 4 0.75

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Data Analysis, Statistics, and Probability

Chapter 6 Data Analysis, Statistics, and Probability Content Strand Description Questions in this content strand assessed students skills in collecting, organizing, reading, representing, and interpreting

### Game Theory and Poker

Game Theory and Poker Jason Swanson April, 2005 Abstract An extremely simplified version of poker is completely solved from a game theoretic standpoint. The actual properties of the optimal solution are

### Chapter 21: The Discounted Utility Model

Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

Education as Degrees and Certificates What is Undergraduate Education? K. P. Mohanan For many people, being educated means attending educational institutions and receiving certificates or degrees. This

### 2. Simple Linear Regression

Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

### Chapter 5: Analysis of The National Education Longitudinal Study (NELS:88)

Chapter 5: Analysis of The National Education Longitudinal Study (NELS:88) Introduction The National Educational Longitudinal Survey (NELS:88) followed students from 8 th grade in 1988 to 10 th grade in

### STUDENTS DIFFICULTIES WITH APPLYING A FAMILIAR FORMULA IN AN UNFAMILIAR CONTEXT

STUDENTS DIFFICULTIES WITH APPLYING A FAMILIAR FORMULA IN AN UNFAMILIAR CONTEXT Maureen Hoch and Tommy Dreyfus Tel Aviv University, Israel This paper discusses problems many tenth grade students have when

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

### ACT Research Explains New ACT Test Writing Scores and Their Relationship to Other Test Scores

ACT Research Explains New ACT Test Writing Scores and Their Relationship to Other Test Scores Wayne J. Camara, Dongmei Li, Deborah J. Harris, Benjamin Andrews, Qing Yi, and Yong He ACT Research Explains

### SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

### IS YOUR DATA WAREHOUSE SUCCESSFUL? Developing a Data Warehouse Process that responds to the needs of the Enterprise.

IS YOUR DATA WAREHOUSE SUCCESSFUL? Developing a Data Warehouse Process that responds to the needs of the Enterprise. Peter R. Welbrock Smith-Hanley Consulting Group Philadelphia, PA ABSTRACT Developing

### 8.7 Exponential Growth and Decay

Section 8.7 Exponential Growth and Decay 847 8.7 Exponential Growth and Decay Exponential Growth Models Recalling the investigations in Section 8.3, we started by developing a formula for discrete compound

### 26. Determinants I. 1. Prehistory

26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.