Mathematics · Year 13

Active learning ideas

Iteration and Fixed Point Iteration

This topic introduces students to fixed-point iteration, a powerful numerical method for finding approximate solutions to equations that cannot be solved algebraically.

• National Curriculum Attainment Targets
• DfE Subject Content for Mathematics: I2 - Solve equations approximately using simple iterative methods, including recurrence relations of the form x(n+1) = g(x(n)).

• 20–30 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Collaborative Problem-Solving
• 25 min · Small Groups

Rearrangement Race

In small groups, students are given an equation like x³ + x - 5 = 0 and must find as many valid rearrangements into the form x = g(x) as possible. They then test each rearrangement with the same starting value to see which ones converge, diverge, or oscillate, fostering a discussion on why some are better than others.

Explain how rearranging an equation f(x) = 0 into the form x = g(x) relates to finding a fixed point.

Facilitation Tip

Encourage groups to compare their findings and hypothesise about the properties of the successful g(x) functions.

What to look for

Use mini-whiteboards for a quick check. Give students an iterative formula and a starting value, and ask them to calculate and display the value of x₁, x₂, and x₃.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 30 min · Pairs

Graphical Iteration Explorer

Using graphing software like GeoGebra or Desmos, students plot y=x and a given y=g(x). They then manually trace the iterative path (from x₀ to g(x₀) on the curve, then horizontally to y=x, then vertically to the curve again) to create their own cobweb or staircase diagram, visually reinforcing the process.

Compare different rearrangements of the same equation and their effect on the convergence of the iterative process.

Facilitation Tip

Provide a worksheet with different functions for g(x) that lead to both convergence and divergence.

What to look for

Set a multi-part exam-style question that requires students to first show a root exists in an interval, then use a given iterative formula to find the root to a specified accuracy, and finally sketch a diagram to illustrate why the iteration converges.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 20 min · Individual

The Convergence Condition

Students are given several iterative formulae and the roots they converge to. They use their differentiation skills to calculate the value of |g'(x)| at the root for each formula and categorise the results based on whether the iteration converges or diverges, helping them to discover the condition |g'(x)| < 1.

Analyse a graphical representation of an iterative process, such as a cobweb or staircase diagram, to determine if it converges or diverges.

Facilitation Tip

Start with a worked example to ensure students are clear on the process of substituting the root into the derivative.

What to look for

Provide students with a problem and a fully worked solution. Ask them to mark their own work, identifying where they made errors and writing a short sentence explaining what they need to remember for next time.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start by graphically linking the root of f(x)=0 to the intersection of y=g(x) and y=x. Model the iterative process on a calculator, emphasising the use of the 'ANS' button to maintain accuracy. Use graphing software to project a cobweb or staircase diagram being built step-by-step to solidify the visual connection.

Students will learn to perform iterative calculations to a required accuracy and will be able to visually interpret the process of convergence or divergence using graphical diagrams.

---

Watch Out for These Misconceptions

• Any rearrangement of f(x) = 0 into x = g(x) will successfully find the root.

  Convergence is not guaranteed. The iterative process only converges to a root 'a' if the gradient of g(x) is sufficiently shallow near the root, specifically, if |g'(a)| < 1. Different rearrangements lead to different g(x) functions with different gradients, so only some will work.

• The result of an iteration is the exact value of the root.

  Iteration provides a sequence of approximations that get progressively closer to the root. The process is stopped when a desired level of accuracy is achieved (e.g., the value is stable to 4 decimal places), but the result is still an approximation, not an exact value.

• A calculator's rounded answer can be used for the next step in the iteration.

  Using a rounded value as the input for the next step introduces rounding errors that can accumulate and affect the final accuracy. Students should always use the full, unrounded answer stored in the calculator's memory for the subsequent iteration.

---

Methods used in this brief

• Collaborative Problem-Solving