Mathematics · Year 13

Active learning ideas

Conditions for Convergence of Iteration

Move beyond solving equations exactly and explore a powerful method for finding highly accurate approximate solutions. We will see how a simple idea from calculus, the tangent line, can be used to build an incredibly fast and efficient root-finding algorithm.

• 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 · Pairs

Visualising Failure Cases

Using dynamic graphing software like GeoGebra or Desmos, students explore scenarios where the Newton-Raphson method fails. They manipulate the initial approximation `x_0` near turning points or between roots to observe divergent, oscillating, or unexpected convergent behaviour.

Explain the role of the derivative, g'(x), in determining the convergence of an iterative formula.

Facilitation Tip

Encourage students to screenshot and annotate their findings for each distinct type of failure.

What to look for

Use mini-whiteboards for students to show their calculated value of `x_1` from a given `f(x)` and `x_0`. This allows for a quick check on their ability to differentiate correctly and apply the formula.

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

Activity 02

• Collaborative Problem-Solving
• 20 min · Small Groups

Iteration Race

In small groups, students solve the same equation using both the Newton-Raphson method and a fixed-point iteration (`x_{n+1} = g(x_n)`). They compare the number of iterations required to achieve a desired accuracy, providing a tangible demonstration of quadratic versus linear convergence.

Justify the condition that |g'(x)| < 1 is required in the vicinity of the root for convergence.

Facilitation Tip

Provide a pre-made spreadsheet template to help students track their iterations and minimise calculation errors.

What to look for

An exam-style question that asks students to perform two iterations of the method, followed by a part that requires them to sketch the function and explain why a different starting value would fail to converge to the desired root.

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

Activity 03

• Collaborative Problem-Solving
• 30 min · Individual

Algorithm Builders

Students implement the Newton-Raphson method as a short program on a graphical calculator or in a simple scripting language like Python. This reinforces the iterative nature of the formula and allows for rapid testing of different functions and starting values.

Analyse why some iterative formulae converge rapidly while others converge slowly or diverge.

Facilitation Tip

Start with a pseudocode structure on the board to guide students who are less confident with programming.

What to look for

Students use a pre-prepared graphing tool (e.g., Desmos) that visualises the Newton-Raphson steps. They can check their manual calculations against the tool and independently explore the effect of changing the initial value.

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

A few notes on teaching this unit

Start by deriving the formula visually, showing the first tangent line intersecting the x-axis to find `x_1`. Use graphing software to repeat the process and show the rapid convergence. After establishing the procedure, dedicate time to exploring failure cases, asking students to predict what will happen if `x_0` is chosen at a turning point before demonstrating.

By the end of this topic, you will be able to confidently apply the Newton-Raphson formula and, more importantly, use a sketch to explain why it works so well and predict when it might fail.

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Watch Out for These Misconceptions

• The Newton-Raphson method will always find a root if one exists.

  The method can fail if the initial approximation is near a turning point (where the tangent is nearly horizontal), causing the next approximation to be very far away. It can also enter an infinite loop or diverge completely.

• The method always converges to the root that is closest to the starting point.

  The tangent line can 'overshoot' a nearby root and converge to a more distant one. The basin of attraction for each root can be complex and non-intuitive, so the closest root is not guaranteed.

• If the method fails, it's because I made a calculation error.

  While calculation errors are possible, failure is often an inherent property of the function and the chosen starting point. Understanding the geometric reasons for failure, such as a horizontal tangent, is a key part of the topic.

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Methods used in this brief

• Collaborative Problem-Solving