Mathematics in Context: Liberal Arts Math Study Guide

Course Overview

Introduction to Mathematics in Context

This course, Mathematics in Context (MGF 1131), is designed for Liberal Arts Math students and covers foundational mathematical concepts and their applications in real-world contexts. The course emphasizes critical thinking, problem solving, and practical mathematical models relevant to personal finance, statistics, probability, voting, and apportionment.

Course Structure and Main Topics

Key Units and Learning Outcomes

• Personal Finance: Understanding and applying mathematical models to personal financial decisions.

• Set Theory: Performing set operations, interpreting Venn diagrams, and analyzing survey data.

• Statistics: Calculating measures of central tendency and dispersion, and interpreting data distributions.

• Probability and Counting: Applying counting principles, permutations, combinations, and probability formulas.

• Voting and Apportionment: Solving problems related to voting methods, fairness criteria, and apportionment methods.

Unit 1: Math of Finance

Simple and Compound Interest

Students learn to calculate interest, rates, present value, and future value using the following formulas:

• Simple Interest Formula: $I = P \times r \times t$

• Future Value (Simple Interest): $FV = P + I$

• Compound Interest Formula: $FV = P \times (1 + \frac{r}{n})^{nt}$

• Effective Annual Yield (EAY): $EAY = (1 + \frac{r}{n})^n - 1$

Example: Calculating the future value of a $1,000 investment at 5% annual interest compounded quarterly for 3 years.

Annuities and Loans

• Future Value of an Annuity: $FV = PMT \times \frac{(1 + r)^n - 1}{r}$

• Present Value of an Annuity: $PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$

Applications include calculating monthly payments, total interest, and evaluating home ownership costs.

Unit 2: Sets and Statistics

Set Theory and Venn Diagrams

Students use set notation and Venn diagrams to analyze relationships between groups and interpret survey data.

• Union: $A \cup B$ (elements in A or B)

• Intersection: $A \cap B$ (elements in both A and B)

• Complement: $A'$ (elements not in A)

Example: Using a Venn diagram to determine how many survey participants belong to multiple categories.

Statistics: Central Tendency and Dispersion

• Mean: $\bar{x} = \frac{\sum x_i}{n}$

• Median: Middle value when data is ordered

• Mode: Most frequent value

• Range: $\text{Range} = \text{Max} - \text{Min}$

• Standard Deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Students also learn about the normal distribution and z-scores:

• 68–95–99.7 Rule: Describes the percentage of data within 1, 2, and 3 standard deviations of the mean.

• Z-score: $z = \frac{x - \mu}{\sigma}$

Unit 3: Counting and Probability

Counting Principles

• Fundamental Counting Principle: Multiply the number of choices at each stage to find total outcomes.

• Permutations: $P(n, r) = \frac{n!}{(n - r)!}$ (order matters)

• Combinations: $C(n, r) = \frac{n!}{r! (n - r)!}$ (order does not matter)

Probability Concepts

• Probability of an event: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Complement: $P( ext{not } E) = 1 - P(E)$

• Union (Or): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Intersection (And): $P(A \cap B) = P(A) \times P(B)$ (if independent)

• Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

Students also learn to convert between odds and probability.

Unit 4: Voting and Apportionment

Voting Methods

• Plurality Method: Candidate with the most votes wins.

• Plurality with Elimination: Candidates are eliminated in rounds until one remains.

• Pairwise Comparison: Candidates are compared head-to-head.

• Borda Count: Points are assigned based on ranking.

Students analyze fairness criteria: Majority, Head-to-Head, Monotonicity, and Irrelevant Alternatives.

Apportionment Methods

• Hamilton’s Method: Assigns seats based on quotas and distributes remaining seats by largest fractional parts.

• Jefferson’s Method: Uses modified divisors to allocate seats.

• Adams’ Method: Similar to Jefferson’s but uses different rounding.

• Webster’s Method: Uses standard rounding to assign seats.

Students learn about apportionment paradoxes (Alabama, Population, New States) and Balinski and Young’s Impossibility Theorem.

Project Assignments: Financial Advisor Project

Applying Mathematics to Real-World Scenarios

• Budget Creation: Analyzing income and expenses to create a balanced budget.

• Car Research: Evaluating vehicle options and financing.

• Housing Options: Comparing renting and buying scenarios.

• Client Portfolio: Compiling financial decisions into a comprehensive plan.

Students use mathematical reasoning to support financial recommendations and reflect on their decisions.

Assessment and Grading

Course Evaluation

• Homework & Practice Assignments: 18%

• Financial Advisor Project: 10%

• Unit Tests (4): Each worth 18%

Final Grade Formula:

$\text{Final Grade} = 0.18(\text{Test 1}) + 0.18(\text{Test 2}) + 0.18(\text{Test 3}) + 0.18(\text{Test 4}) + 0.18(\text{Average of all Homework and Practice Assignments}) + 0.10(\text{Average of all Financial Advisor Project Assignments})$

Scratch paper submission is required for unit tests and counts toward the test grade.

Tips for Student Success

Strategies for Effective Learning

• Stay current with assignments and refer to the pacing calendar.

• Engage with class notes, videos, and worked-out solutions.

• Ask questions and attend virtual support hours.

• Practice regularly and review material before tests.

• Prepare and submit scratch paper showing all work.

• Use available resources: Canvas, MyLab Math, Learning Commons, and Library.

• Manage time proactively and believe in your ability to improve.

Academic Integrity and Course Policies

Expectations and Support

• All work must be original unless collaboration is authorized.

• AI tools may be used for project assignments but not for graded assessments.

• Attendance is based on active participation.

• Withdrawal and makeup policies are outlined in the syllabus.

• Support services are available for students with disabilities.

Summary Table: Major Course Topics

Main Topic	Key Concepts	Example Formula
Finance	Simple/Compound Interest, Annuities	$I = P \times r \times t$
Sets	Union, Intersection, Complement	$A \cup B$, $A \cap B$, $A'$
Statistics	Mean, Median, Mode, Range, SD	$\bar{x} = \frac{\sum x_i}{n}$
Probability	Counting, Permutations, Combinations	$P(n, r) = \frac{n!}{(n - r)!}$
Voting	Plurality, Borda Count, Fairness Criteria	N/A
Apportionment	Hamilton, Jefferson, Adams, Webster	N/A

Additional info: The course covers all major Liberal Arts Math topics, including sets, logic, numeration systems, real numbers, algebra review, geometry, voting, apportionment, and graph theory, as outlined in the provided chapter titles.