Mathematics as Algorithms Using Python

Objective:

To introduce students to fundamental algorithms using Python, showing how these concepts can be applied to solve real-world math problems. This course emphasizes problem-solving skills, logical thinking, and computational fluency.

Platform:

Web-based Python (using platforms like Replit, Trinket, or Google Colab)

Weekly Breakdown:

Week 1: Introduction to Algorithms and Python Basics

Objective: Understand what algorithms are and their importance in problem-solving.
Content:

Introduction to the course and its objectives.
Overview of Python basics: variables, data types, and simple operations.
Simple Python exercises using web platforms.

Activity: Write a basic program that adds, subtracts, multiplies, and divides numbers.

Week 2: Understanding Sorting Algorithms (Bubble Sort)

Objective: Learn to sort lists of numbers using the Bubble Sort algorithm.
Content:

Introduction to sorting and why it's useful.
Step-by-step guide to Bubble Sort.
Implementing Bubble Sort in Python.

Activity: Sort a list of test scores using Bubble Sort.

Week 3: Solving Selection Problems (Greedy Algorithms)

Objective: Use greedy algorithms to solve optimization problems.
Content:

Understanding greedy algorithms and how they work.
Example: Finding the minimum number of coins needed for change.
Implementing the coin change problem in Python.

Activity: Write a program that finds the optimal way to give change.

Week 4: Introduction to Knapsack Problems (Dynamic Programming)

Objective: Solve optimization problems using the knapsack algorithm.
Content:

Understanding the knapsack problem in real life.
Introduction to dynamic programming basics.
Implementing a simple knapsack problem in Python.

Activity: Plan a trip by selecting items to pack in a limited space suitcase.

Week 5: Scheduling Algorithms (Task Scheduling)

Objective: Learn to schedule tasks to maximize efficiency.
Content:

Introduction to task scheduling and real-world examples.
Implementing task scheduling algorithms in Python.
Example: Scheduling study and playtime effectively.

Activity: Create a study schedule that fits within a daily time limit.

Week 6: Graph Algorithms (Shortest Path)

Objective: Find the shortest route between locations using algorithms.
Content:

Introduction to graphs and paths.
Implementing a simple shortest path algorithm (Dijkstra's).
Real-world application: Planning the shortest route for errands.

Activity: Create a program that finds the shortest route between points.

Week 7: Optimization Algorithms (Subset Sum)

Objective: Solve subset sum problems using backtracking.
Content:

Explanation of subset sum and its applications.
Step-by-step coding of a subset sum solution in Python.
Applying it to select the best combination of items under constraints.

Activity: Plan a shopping list to maximize items within a budget.

Week 8: Probability and Simulation (Random Walks)

Objective: Understand randomness and simulation using Python.
Content:

Introduction to random number generation and simulations.
Coding a random walk simulation.
Exploring how simulations can solve probabilistic problems.

Activity: Simulate a random walk and visualize the results.

Week 9: Game Algorithms (Minimax for Decision Making)

Objective: Use algorithms to make decisions in games.
Content:

Introduction to Minimax and its use in decision-making.
Implementing a simple game decision algorithm.
Applying it to a game scenario like tic-tac-toe.

Activity: Build a basic tic-tac-toe game with decision-making.

Week 10: Recursion and Recursive Algorithms

Objective: Learn about recursion and how it simplifies problem-solving.
Content:

Introduction to recursion with simple examples.
Applying recursion to solve problems like factorials and Fibonacci.
Understanding recursive problem-solving strategies.

Activity: Write a recursive function to solve a problem like finding the greatest common divisor.

Week 11: Sorting Revisited (Quick Sort and Merge Sort)

Objective: Explore advanced sorting algorithms and their efficiencies.
Content:

Differences between Bubble, Quick, and Merge Sort.
Implementing Quick Sort and Merge Sort in Python.
Comparing sorting speeds and efficiency.

Activity: Compare sorting times for different algorithms using sample data.

Week 12: Project Week – Building a Real-World Application

Objective: Consolidate knowledge by applying learned algorithms to a real-world project.
Content:

Students will choose a problem they can solve using one or more algorithms learned.
Guidance on project structure, implementation, and presentation.
Coding the project and testing.

Activity: Present the project and explain the algorithm used.

Tools and Resources:

Web Platforms: Replit, Trinket, Google Colab (no installation needed).
Language: Python 3
Materials: Worksheets, code snippets, and guided video tutorials.

Outcome:

By the end of this program, students will be able to understand and implement algorithms in Python to solve complex math problems, develop a foundation in computational thinking, and appreciate the practical applications of mathematics.