Lesson Plan: Introduction to Mathematical Programming
Grade Level: High School/Undergraduate
Duration: 90 minutes
Prerequisites: Basic algebra, Python syntax (or similar language，oy430a7.com)
Learning Objectives
- Understand how to translate mathematical problems into code.
- Implement algorithms for solving equations, optimization, and simulations.
- Debug and analyze code output for mathematical correctness.
1. Warm-Up: Mathematical Problems as Code Challenges (15 mins)
Activity:
- Present a simple problem: "Find the roots of the quadratic equation ax2
- +bx+c=0."（oy430a6.com）
- Break into groups to write pseudocode before coding.
Solution (Python Example):
python
import math def quadratic_roots(a, b, c): discriminant = b**2 - 4*a*c if discriminant < 0: return "No real roots" root1 = (-b + math.sqrt(discriminant)) / (2*a) root2 = (-b - math.sqrt(discriminant)) / (2*a) return root1, root2 # Example usageprint(quadratic_roots(1, -3, 2)) # Output: (2.0, 1.0)
Discussion:
- How does the code handle edge cases (e.g., negative discriminant)?
- What if a = 0? (Introduce error ，oy430a5.cc，handling.)
2. Core Concept: Algorithms for Mathematical Problems (30 mins)
A. Optimization: Maximizing a Function
Example: Find the maximum of f(x)=−x2
+4x using a brute-force search.
python
def find_max(start, end, step): max_value = float('-inf') best_x = None for x in [start + i*step for i in range(int((end-start)/step)+1)]: y = -x**2 + 4*x if y > max_value: max_value = y best_x = x return best_x, max_value print(find_max(0, 5, 0.01)) # Output: (2.0, 4.0)
Extensions:
- Compare with calculus-based solution (x=−b/(2a)).
- Introduce gradient descent for more efficient optimization.
B. Simulation: Probability and Randomness
Example: Estimate π using Monte Carlo simulation.
python
import random def estimate_pi(num_samples): inside_circle = 0 for _ in range(num_samples): x, y = random.random(), random.random(oy430a4.cc) if x**2 + y**2 <= 1: inside_circle += 1 return 4 * inside_circle / num_samples print(estimate_pi(100000)) # Output: ~3.14
Key Takeaway: How randomness approximates deterministic values.
3. Hands-On Activity: Group Coding Challenge (25 mins)
Task:
- Write a program to calculate the first n Fibonacci numbers using:
- A recursive function (inefficient，oy430a8.com).
- A loop (efficient).
- Compare runtime for large n (e.g., n = 40).
Solution (Loop Version):
python
def fibonacci(n): a, b = 0, 1 for _ in range(n): print(a, end=" ") a, b = b, a + b fibonacci(10) # Output: 0 1 1 2 3 5 8 13 21 34
Discussion:
- Why is recursion slow here? (Introduce time complexity.oy430a3.cc)
- How could you optimize further (e.g., oy430a9.com，memoization)?
4. Debugging & Validation (15 mins)
Activity:
- Present a buggy code snippet (e.g., incorrect factorial function，oy430a10.com).
- Students debug in pairs and explain the fix.
Buggy Example:
python
def factorial(n): result = 1 for i in range(n): # Bug: Should be range(1, n+1) result *= i return result print(factorial(5)) # Wrong output: 0 (due to i=0 oy430a2.cc，multiplication)
Fix:
python
def factorial(n): result = 1 for i in range(1, n+1): # Corrected result *= i return result
5. Assessment & Homework (5 mins)
Exit Ticket:
- Write a function to check if a number is prime.
- Bonus: Optimize it to skip even divisors after checking 2.
Homework:
- Solve Project Euler Problem 1 (sum of multiples of 3 or 5 below 1000，51yf3.cc).
- Research: How do real-world mathematicians use programming (e.g., cryptography,51yf5.cc， climate modeling)?
Teaching Tips
- Start Simple: Use familiar math (quadratics,51yf6.com， Fibonacci) before advanced topics.
- Visualize: Use tools like Desmos or Matplotlib to plot functions/results.
- Error Handling: Teach try-except blocks early for robust code.
- Real-World Links: Show how Python libraries (NumPy, 51yf4.ccSciPy) solve complex problems.
This lesson balances theory and practice, empowering students to think mathematically and computationally. Let me know if you'd like expansions on specific topics!
