Solving Equations Through Iterative Methods
Year 9 Mathematics Cambridge IGCSE 50-minute lesson
WALT - We Are Learning To
Understand iteration as repeated calculations Use iterative processes to solve simple equations Interpret results from iterative sequences Develop problem-solving skills using iteration
What is Iteration?
Think about these examples: Counting in steps: 2, 4, 6, 8... Following a recipe repeatedly GPS recalculating your route What do they have in common?
Iteration in Mathematics
Iteration = repeated process with a rule Start with an initial value (x₀) Apply the same formula repeatedly Each result becomes the input for the next step Continue until values stabilize or converge
Hands-On Activity: Square Root Iteration
Formula: x_{n+1} = ½(x_n + 4/x_n) Start with x₀ = 2 Calculate x₁, x₂, x₃, x₄ Record your results in a table What do you notice about the values?
Convergence vs Divergence
{"left":"Convergence: Values get closer to a fixed point\nExample: 2.5 → 2.05 → 2.0006 → 2.000...","right":"Divergence: Values move away or oscillate\nExample: 1 → 3 → 9 → 81 → 6561..."}
Physical Iteration Game
Work in groups of 3-4 students Draw cards with iteration rules Example: 'Multiply by 0.5 and add 1' Take turns applying the rule Record your sequence on the poster Discuss: Does your sequence converge?
Success Criteria & Next Steps
Can you explain iteration in your own words? Can you perform iterative calculations? Can you recognize convergence vs divergence? Extension: Research iteration in technology Next lesson: Graphical iteration methods