Study case: using iteration from a rearranged formula In this GCSE Iteration Rearranging Formula Problem Tutor, I use one clear study case to show how iteration works as a repeated process. The problem is: use the iteration formula x_{n+1} = sqrt(10 - x_n), starting with x_0 = 3, to estimate a solution. I like this example because students can follow the repeated substitution, but they also need to understand why values settle towards an answer. The process gives values close to 2.7. Before solving, I ask the student to identify the current value and the next value. x_n is the value we have now, and x_{n+1} is the value we calculate next. This notation can be intimidating, so I explain it in ordinary language: put the current answer into the right-hand side, then the result becomes the next answer. The formula is not solved in one step; it is used repeatedly. Method 1 is direct substitution. We start with x_0 = 3. Then x_1 = sqrt(10 - 3) = sqrt(7), which is about 2.646. Next, x_2 = sqrt(10 - 2.646), which is about 2.712. Then x_3 = sqrt(10 - 2.712), which is about 2.700. The values are settling, so the solution is approximately 2.70. I teach students to keep enough decimal places during working so rounding does not distort the result too early. Method 2 is calculator table organisation. I ask the student to create a small table with columns for n and x_n. This table might show x_0 = 3, x_1 = 2.646, x_2 = 2.712, x_3 = 2.700 and x_4 = 2.702. A table helps students see the pattern. It also makes the answer easier to check because the values should begin to stabilise. If the values jump around wildly or become impossible, the student should question the formula or the input. Method 3 is checking with the original equation. If the iteration is based on x = sqrt(10 - x), then squaring gives x^2 = 10 - x, so x^2 + x - 10 = 0. If x is about 2.70, then x^2 + x - 10 is about 7.29 + 2.70 - 10, which is close to zero. This check shows that the iteration has led to a sensible approximate solution. I use this method to connect iteration to algebra rather than treating it as a calculator-only process. After the methods, I compare them with the student. Direct substitution teaches the mechanics of the iteration. A table organises the work and prevents lost values. Substitution back into the equation checks whether the approximation is sensible. I want students to see iteration as a structured numerical method, not as pressing buttons without understanding. The most common mistakes are using x_0 again instead of using the latest value, rounding too early, typing the formula incorrectly into the calculator, and not giving the answer to the required accuracy. Some students also confuse x_n and x_{n+1}. I slow this down by asking them to say: this value becomes the next input. That sentence helps the process become clear. This topic is important because GCSE iteration questions often test formula use, calculator accuracy and approximation. Students may be asked to show the first few iterations, explain why values are converging, or give a solution to a specified number of decimal places. Clear layout is essential. If the work is messy, even a correct method can become hard to follow. A useful extension is to derive the iteration formula from an equation. For example, from x^2 + x - 10 = 0, one rearrangement is x = sqrt(10 - x). Another rearrangement may behave differently. This helps students understand that rearranging is not always neutral in iteration; some formulae converge better than others. In one-to-one tutoring, I would finish by giving the student a similar iteration formula and asking them to complete a table independently. I would then ask them to explain how they know when the answer has settled. This builds confidence and prevents iteration from feeling like a mysterious calculator trick. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit GCSE and IGCSE Maths tutoring: https://www.mastermathstutoring.co.uk/gcse-igcse. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.