Welcome to GAMES! A Gentle Approach to Math, Excel, and Stats: Course 1

Who is this course for?

This course is for a diverse set of learners in business, economics, and other social sciences including (a) those who may lack a strong enough math and statistics background from high school, (b) learners seeking to refresh their math and statistics knowledge in preparation for upper years in university and college, (c) learners wishing to re-skill or up-skill by switching majors or adding a minor and lacking the necessary math and statistics tools.

What does this course cover?

The course has two segments.

Segment 1 covers precalculus algebra. These topics are necessary for performing mathematical calculations and higher-level mathematics. It can be useful to think of algebra as a language with its own definitions and rules. Topics are introduced assuming the learner has forgotten all their high school math. We begin by relating algebraic expressions to real world examples from the social sciences. These examples demonstrate the utility of mathematics to motivate the learner as more technical topics are introduced. Sets and real numbers are explored, followed by exponents, factors, and fractions. These mathematical concepts are illustrated using two-dimensional graphs, and their applications are demonstrated with real-world examples whenever possible. Functions and summation notation are important tools for relating real-world phenomena to mathematics in a way that is useful for analysis and simplifies communication.

Segment 2 introduces differential calculus which is perhaps the most commonly used mathematical concept in business and the social sciences. We begin by introducing the concept of limits: the value of a function as the input variable moves closer and closer to some particular number. Limits help understand the concept of differentiation which is in turn used to calculate rates of change for expressions more complicated than a straight line. We develop the important idea that on a graph, differentiation equates to the slope of a curve. We then explore three useful applications of differential calculus: implicit differentiation, linear approximation, and optimization.

Each module relies on at least some of the topics covered in a previous module, so we suggest that the learner work through each module in sequence. A learner who has not mastered the content in part 1 is likely to be challenged to develop even a basic understanding of the content in part 2.

Learning objectives

By following the lectures and attempting the embedded and practice questions, the learner should be able to:

• Model real-world phenomena as an equation and inequality.
• Demonstrate an understanding of the basics of limits and summation notation.
• Create a two-dimensional diagram to represent the mathematical relationship between two variables, both by hand and using Excel or another spreadsheet program.
• Identify all points of interest on a diagram including the axes labels, slopes, and intercepts.
• Identify expressions that are implicit functions and apply implicit differentiation to find the slope of these expressions.
• Use linear approximation to approximate the level of a variable and its rate of change.
• Convert a word problem into a mathematical optimization problem and solve this optimization problem.

Course Resources

Students have access to the following course resources:

• Close-captioned lecture videos
• Embedded questions (with solutions) in the lecture videos to encourage retention and provide immediate feedback
• Full lecture transcripts
• Blank and annotated lecture slides
• Practice problems, including some computations with spreadsheet software such as Excel / Google sheets (provided in both .xls and .ods formats)
• Solutions to the practice problems, including Excel computations (provided in both .xls and .ods formats)

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) https://creativecommons.org/licenses/by-nc-nd/4.0/ Unless otherwise noted, all content in this video was created by Catherine Pfaff, Sumon Majumdar, and Robert J. McKeown. You are free to copy and share this material in any format, but you must give appropriate credit to the authors. This project is made possible with funding by the Government of Ontario and through eCampusOntario’s support of the Virtual Learning Strategy. To learn more about the Virtual Learning Strategy visit: https://vls.ecampusontario.ca/.

Watch Our Introduction to GAMES Video

Acknowledgements

There are many who helped us directly and indirectly with developing the two courses. The three of us have been teaching segments of the material covered here in various courses that we have taught over the years. We would like thank the many, many students in our courses who have indirectly helped us develop, revise and fine-tune the material and sharpen our teaching.

We would like to thank Alan Ableson, Andrew McEachern, William Nelson, Andrew Skelton and Victoria Sytsma for their subject-matter expertise, Randy Ellis for allowing Catherine use his Lightboard studio, and Robert Winkler, Yelin Su and the Teaching Commons at York University for reviewing and user-testing the accessibility of the courses.

We had many students at Queen’s and York University who reviewed the videos and the embedded questions, helped us generate examples, edited and formatted the videos and contributed in myriad ways, all of which have immensely enhanced the quality of the courses. We would like to thank Rebecca Carter and Vladimir Fenenko for their research and examples in our teaching material. Sean Steele and Alexis Blair Hamilton were essential in creating the interactive videos and transcripts with a tight deadline. Sebastian Vaillancourt, Haozhi (Howie) Hong, and Thurston Han edited many of our videos. Azeezah Jafry, Constance Anson, Diana Liu Xu, Rajbalinder Singh Ghatoura, Tao Tang, and Thashvin Ramnauth provided actionable feedback on our interactive videos.

Finally, we would like to thank Aya Javid at York University and Jenna Dijkema at Queen’s University for handling administrative matters related to the creation of the courses.

This project is made possible with funding by the Government of Ontario and through eCampusOntario’s support of the Virtual Learning Strategy. To learn more about the Virtual Learning Strategy visit: https://vls.ecampusontario.ca/.

Modules

Module 1: Variables and Algebraic Expressions in the Social Sciences

Most areas in the social sciences now use algebraic expressions to succinctly model and analyze real-life situations. In this module, using examples from a variety of contexts, we introduce the idea of depicting a given real-world situation using an algebraic expression involving some essential variables. Subsequent analysis of such expressions, using mathematical tools (together with spreadsheet software such as Excel or Google Sheets) can help develop deep insights into the problem and the impact of policies and other changes. Examples in this module are drawn from politics, business, geography, sports analytics and modelling the spread of infections.

By the end of the module, the student should be able to:

• Formulate some common situations using an algebraic expression / formula
• Analyze an algebraic expression to answer questions relevant to the social sciences
• Use Excel or Google Sheets to computationally analyze and compare between different real-world situations and policies

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures, and lecture transcripts.

Lectures

• 1.1 Introduction to Working with Variables
• 1.2 Example from Politics
• 1.3 Example from Politics Continued
• 1.4 Excel Data Analysis: Using Formulas to Rank Athletes
• 1.5 Excel Data Analysis 2: Using Formulas to Rank Athletes Continued
• 1.6 Example from Geography
• 1.7 Modelling the Spread of a Pandemic 1
• 1.8 Modelling the Spread of a Pandemic 2
• 1.9 Excel: Modelling the Spread of a Pandemic 1
• 1.10 Excel: Impact of Policies on the Spread of a Pandemic 1
• 1.11 Excel: Impact of Policies on the Spread of a Pandemic 2

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 2: Sets & Numbering Systems

This module introduces a simple but essential element of most mathematical problems: sets. Before any function or algebraic expression is defined, it is important to identify the set over which the definition holds. Starting with real numbers and integers, the module develops the various ways of writing sets, both finite and infinite. The latter half of the module focuses on the essential concepts in combining sets, namely their union and intersection.

By the end of the module, the student should be able to:

• Succinctly write a set
• Develop an understanding of finite and infinite sets
• Develop the union and intersection of given sets

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 2.1 Real Numbers, Integers, & Their Notation
• 2.2 Integer Examples/Nonexamples
• 2.3 Some Ways to Write a Set
• 2.4 Inequalities & Finite Sets: Part 1
• 2.5 Inequalities & Finite Sets: Part 2
• 2.6 Writing Infinite Sets with Patterns
• 2.7 Set Overlaps: Intersections
• 2.8 Combining Sets: Unions
• 2.9 Intersections of Intervals
• 2.10 Unions of Intervals

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 3: Exponents, Factors and Fractions

This module continues to develop your ability to manipulate algebraic expressions by introducing more advanced operators such as Exponents and understanding their interaction with real numbers and fractions. Fractions are common in the social sciences and business, and successfully manipulating their properties is a prerequisite for your future success. Exponents act as a more succinct instruction to multiply, while factors allow for solving many common algebraic expressions with more than one solution. While this module is predominantly a technical one, the usefulness of the skills cannot be overestimated. An application applying fractions to an ESG (environmental, social, governance) investment portfolio is considered.

By the end of this module, the student should be able to:

• Manipulate exponents with a common base.
• Memorize the three most common factor identities and apply them to solving common algebraic expressions.
• Memorize the rules for fractions and apply them to finding common denominators and understanding investment portfolios.

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures, and lecture transcripts.

Lectures

• 3.1 Adding and Subtracting Exponents
• 3.2 Multiplying and Understanding Exponents
• 3.3 Exponents in Practice: Savings and Interest Rates
• 3.4 Algebra: BEDMAS and FOIL
• 3.5 Terms, Factors and Factoring
• 3.6 Algebra Example and Factoring
• 3.7 Factoring with Identities
• 3.8 Factoring More Advanced Expressions
• 3.9 The Rules for Fractions
• 3.10 Common Denominators and Simplifying Fractions
• 3.11 Fractions in Practice: Investment Portfolios

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 4: Fractions and Systems

The module begins by considering fractions in the exponent which combines two skills developed previously in module 3. The concept of a function is next introduced. A function is a useful concept that is analogous to a machine that takes inputs to produce outputs. Building on simple examples like a vending machine, we then introduce compositions of functions - functions within other functions. We consider an application from economics using a Laffer Curve which combines tax policy with labor incentives to demonstrate optimal (revenue maximizing) taxation policy. The remainder of the module focuses on an important technical skill: manipulating a system of equations. A variety of techniques to solve systems of equations are presented including the substitution and elimination methods.

By the end of this module, the student should be able to:

• Apply the rules of fractions to fractions in the exponent.
• Define a function and identify real-world examples of functions.
• Solve composition of functions models such as the Laffer Curve by hand and with Excel or another spreadsheet program.
• Solve a system of equations using both the substitution and elimination methods.

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures, and lecture transcripts.

Lectures

• 4.1 Fractions in the Exponent 1
• 4.2 Fractions in the Exponent 2
• 4.3 Functions: an Introduction
• 4.4 Functions: Notation and Language
• 4.5 Compositions of Functions
• 4.6 Excel: Composition of Functions
• 4.7 Compositions of Functions: Examples I
• 4.8 Compositions of Functions: Examples 2
• 4.9 Systems of Equations
• 4.10 Elimination Method
• 4.11 System of Equations: Examples 1
• 4.12 System of Equations: Examples 2
• 4.13 System of Equations: Examples 3

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 5: Graphs and the Euclidean Plane

In this module, we focus on the graphical representation of two variables. Two-dimensional representations are extremely powerful methods for communicating findings. Even in the most sophisticated and highly regarded academic paper, one can usually find a two-dimensional plot.

By the end of this module, the learner should be able to

• identify points in Euclidean space
• draw mathematical functions on a two-dimensional graph
• contrast a linear function with a nonlinear function

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 5.1 Euclidean (or Coordinate) Plane R^2
• 5.2 The Points of a Line Are a Set in R^2
• 5.3 Graphs of Lines are Graphs of Linear Functions
• 5.4 Linear Word Problem Example
• 5.5 Changing an Equation to a Function
• 5.6 Graph of Functions Second Example
• 5.7 Example Connecting Word Problems, Functions, and Graphs
• 5.8 Distance in R^2
• 5.9 Fun Example: Voronoi Diagrams
• 5.10 x-Intercepts and y-Intercepts

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 6: Functions in Practice

A more mathematical definition of functions is necessary to create useful models of the real world that can be algebraically rearranged to produce important insights into human behaviour. We introduce discrete and continuous functions while explaining the concept of the domain and range. Also included is a simple test to determine whether a mathematical relationship is a function, called the vertical line test. This understanding of functions and inverse functions is applied to models from economics and business. Models are solved by hand and using Excel or another spreadsheet program.

By the end of this module, the student should be able to:

• Define a function in mathematical terms and identify mathematical relationships as either a function or not a function.
• Use set notation to limit the domain and range of a function or mathematical relationship.
• Define the conditions under which a specific function has an inverse function.
• Manipulate and solve models with composite functions by hand and with Excel or another spreadsheet program.

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 6.1 Discrete Functions: Domain and Range
• 6.2 Continuous Functions: Domain and Range
• 6.3 The Vertical Line Test
• 6.4 One-to-One Functions
• 6.5 Inverse Functions
• 6.6 Functions in Practice: Inverse Demand
• 6.7 Excel: Graphing a Function and its Inverse
• 6.8 Functions in Practice: Supply and Demand
• 6.9 Excel: Graphing Supply and Demand
• 6.10 Excel: Graphing a Linear Revenue and Quadratic Cost Function

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 7: Summation Notation, Series and Sequences

The summation ∑ notation is one of the most widely used notations in the social sciences. However, many students often struggle with it. In this module, we focus on developing familiarity with the meaning and usage of the summation notation. Furthermore, we introduce students to two of the most common sums: the arithmetic and geometric series. As an application, we develop the formula for the valuation of annuities and perpetuities. The module also covers doing summations and using some in-built functions in Excel or Google Sheets, and the limits of some common sequences.

By the end of the module, the student should be able to:

• Interpret summation notation and develop summation notation for a given situation
• Sum an arithmetic or geometric series
• Use in-built functions in Excel or Google Sheets for some common mathematical computations

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures, and lecture transcripts.

Lectures

• 7.1 Summation Notation
• 7.2 Examples of Summation Notation
• 7.3 Summation Notation in Statistics
• 7.4 More Applications
• 7.5 Summation Problems and Applications
• 7.6 Excel Data Analysis: Doing Summations
• 7.7 Arithmetic Series 1: Gauss' Formula
• 7.8 Arithmetic Series 2: Gauss' Formula and Applications
• 7.9 Arithmetic Series 3: More Applications
• 7.10 Geometric Series 1: Basic formula
• 7.11 Geometric Series 2: Applications
• 7.12 Sequences and their Limits
• 7.13 More Limits of Sequences
• 7.14 A Few More Limits of Sequences

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Course 1 Term Test

This test covers modules 1-7. Try to complete the test in less than 90 minutes without looking at your notes or the internet - just a piece of paper and a calculator. After completing the test on your own with no outside help, compare your answers to those in the solution file.

Course 1 Term Test

Module 8: Limits

This module introduces students to limits, the idea of investigating where a function or process is headed towards as one continues to change a variable in a particular way. Understanding limits is central to understanding the subsequent important topic of differentiation. The concept is also useful for social scientists in understanding where a social process is headed towards and where it may end up. The module helps students develop the basic idea of limits in general, and also the tools for computing limits of the most common types of functions.

By the end of the module, the student should be able to:

• Develop an understanding of the idea of the limit of a function.
• Take limits of some common types of functions.
• Understand and utilize some properties of limits.

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 8.1 Meaning of Limit Notation
• 8.2 Left-Limit vs Right-Limit
• 8.3 Limits of Quotient Functions
• 8.4 Limits of Exponential Function and Natural Logarithm
• 8.5 Taking Limits is a "Linear Operation"
• 8.6 Example of Applying Limit Properties
• 8.7 Direct Substitution Property for Limits of Polynomials and Rational Functions
• 8.8 Example of Applying Limit Properties
• 8.9 Some More Useful Limit Properties: Produce, Quotient, & Power Rules
• 8.10 Some More Useful Limit Properties: Roots
• 8.11 Limit when f(x) not defined at x=a: Part 1
• 8.12 Limit when f(x) not defined at x=a: Part 2
• 8.13 Infinity - Infinity Could Be Zero

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 9: Introduction to Derivatives

This module introduces one of the most commonly used mathematical concepts in business, economics and other social science subjects, namely the concept of a derivative. Starting from the rate of change of a line and average velocity, it moves to the idea of instantaneous velocity and tangent of a curve to establish the concept of derivative of a function. The module then introduces the formal definition of a derivative and uses the definition to establish derivatives of some simple functions. It also exposes students to the related idea of non-differentiability.

By the end of the module, the student should be able to:

• Develop an understanding of the idea of derivative of a function.
• Know the formal definition of a derivative and be able to use the definition
• Understand differentiability and non-differentiability of a function

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 9.1 Slope as Rate of Change for Lines
• 9.2 Slope and Average Rate of Change for Functions
• 9.3 Example of Rate of Change from Limited Data
• 9.4 From Average Velocity to Instantaneous Velocity
• 9.5 Approximate Rate of Change to Rate of Change is Taking a Limit
• 9.6 Instantaneous Rate of Change Slope of Tangent Line Derivative
• 9.7 Introduction to Tangent Lines (Useful for Approximation)
• 9.8 The Derivative Function
• 9.9 Visual Example for the Derivative Function
• 9.10 Derivative Language: "With Respect To"
• 9.11 Using Derivative Definition to Compute Derivative of Constant
• 9.12 Computing Derivative of g(x)=x^2 from Definition
• 9.13 Differentiability
• 9.14 Example of Showing a Function's Not Differentiable
• 9.15 Not "Differentiable at a" Pictures

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 10: Derivative Computations

Additional rules are developed that allow the learner to find the derivative of more complicated expressions and functions. The product rule simplifies the process of taking the derivative when two or more functions are multiplied with each other. The chain rule allows the learner to take the derivative even when functions are embedded in other functions, called composite functions. Combining the two rules allows the learner to find the derivative even for very long and complicated expressions. We conclude with an introduction of higher-order derivatives such as the second derivative which is the rate of change of the rate of change.

By the end of this module, the student should be able to:

• Apply the product rule and chain rule to find the derivative of simple algebraic expressions and functions.
• Combine the product rule and chain rule to find the derivative of more complex expressions and functions.
• Explain how the second derivative is derivative of the derivative or the rate of change in the rate of change.

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 10.1 Computing Derivatives: First Tools
• 10.2 Computing Derivatives: Polynomials
• 10.3 Differentiation with "Funny Powers" Example
• 10.4 A Few (More) Good Derivatives to Know
• 10.5 Where Product Rule Comes From
• 10.6 Chain Rule by Example
• 10.7 Practice Identifying "Inside" and "Outside" In Chain Rule
• 10.8 Combining Product Rule and Chain Rule Example
• 10.9 Combining Product Rule and Chain Rule Example 2
• 10.10 Higher Derivatives Introduced

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 11: Applied Differential Calculus

The tools of single variable calculus are extended to implicit functions where the value of y depends implicitly on x. A variation on the previous methods of computing derivatives of implicit differentiation is introduced. Implicit differentiation allows for analyzing algebraic expressions that are too complicated for one variable to be arranged and isolated on a single side of an expression. To start, the technique is applied to the equation of a circle followed by more complicated expressions. Next, we introduce linear approximations and how derivatives can be used to approximate change in a variable of interest. We conclude with an example from finance looking at the price of fixed income bonds using Excel or another spreadsheet program.

By the end of this module, the student should be able to:

• Identify an implicit function, take its derivative, and find the slope.
• Graph the implicit function on a diagram and identify the tangent line (slope).
• Use a linear approximation to estimate the change in one variable from the change in another variable.
• Demonstrate your understanding of linear approximation by analyzing the change in the price of a bond from a change in interest rates using Excel or another spreadsheet program.

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 11.1 Implicit Functions
• 11.2 Implicit Differential
• 11.3 Implicit Differential Examples 1
• 11.4 Implicit Differential Examples 2
• 11.5 A Two-Period Model of Consumption
• 11.6 Second Implicit Differential
• 11.7 Implicit Differential: Functions in an Expression 1
• 11.8 Implicit Differential: Functions in an Expression 2
• 11.9 Linear Approximations
• 11.10 Bond Prices and Interest Rates Examples I
• 11.11 Bond Prices and Interest Rates Examples 2
• 11.12 Excel: Bond Prices and Interest Rates Example I
• 11.13 Excel: Bond Prices and Interest Rates Example 2

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 12: Optimization

Finding the optimal use of a scarce resource is fundamental to making smart decisions. The decision-making process becomes more challenging when the choice is made over a continuous set such as how much to spend, what price to charge, or how much product to create. Business, economics, and the broader social sciences often encounter these types of quantitative problems. We introduce methods to find extreme values: the maximum and minimum of a function over a continuous domain. Real-world examples are explored by hand and with Excel or another spreadsheet program. The intuition underlying single variable optimization can be extended to multiple variables.

By the end of this module, the student should be able to:

• Determine if a global maximum or global minimum exists and to identify where this point occurs using Excel or another spreadsheet program,
• Differentiate an equation for the purpose of optimization, solve for its critical points and evaluate whether the critical point is a maximum, a minimum, or neither.
• Apply the above optimization techniques to common problems from economics and business by hand and using Excel or another spreadsheet program.

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures and lecture transcripts.

Lectures

• 12.1 Extreme Points
• 12.2 Maximums and Minimums
• 12.3 Optimization with Differential Calculus
• 12.4 Critical Points
• 12.5 Optimization Examples 1
• 12.6 Optimization Examples 2
• 12.7 How to Find Global Extrema
• 12.8 First Derivative Test
• 12.9 First Derivative Test Examples
• 12.10 Second Condition Test
• 12.11 Second Condition Test Examples
• 12.12 Excel: Solving an Optimization Problem

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Course 1 Final Exam

This exam covers modules 1-12 with particular emphasis on modules 8-12. Try to complete the test in less than 120 minutes without looking at your notes or the internet - just a piece of paper and a calculator. After completing the test on your own with no outside help, compare your answers to those in the solution file.

Course 1 Final Exam