OT2: Mathematical problem solving: Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions, including those obtained using numerical methods.

The idea

After obtaining a solution, you must judge how reliable it is. For numerical answers, consider the method used, the number of iterations, and the effect of rounding. A reasoned estimate can come from a rough sketch, a simpler model, or bounds. For example, if a numerical root is 1.4142, you might estimate that the true value lies between 1.414 and 1.415. Limitations include slow convergence, sensitivity to initial guesses, or assumptions in the model. You should comment on whether the accuracy is sufficient for the problem’s context and identify any factors that could make the solution unreliable. This evaluative step is a key part of mathematical problem solving.

What earns the marks

• Makes a reasoned estimate of a solution using a sketch, bounds, or a simpler calculation.
• Compares a numerical solution with an estimate to judge its plausibility.
• Identifies sources of error, such as rounding, truncation, or iteration count.
• Discusses limitations of the method, including convergence issues or sensitivity to starting values.
• States whether the solution is accurate enough for the given context and justifies that judgement.

Common mistakes and how to put them right

• Accepting a numerical answer without checking it against a rough estimate.

  Put it right: Perform a sanity check using a graph or simple bounds.

• Ignoring the effect of rounding during intermediate steps.

  Put it right: Keep extra digits in working and round only at the final stage.

• Claiming a solution is exact when it is approximate.

  Put it right: Use language such as ‘approximately’ and state the level of accuracy.

Exam tips

• Include a rough sketch or a quick calculation to support your estimate.
• Comment explicitly on the accuracy of your answer, referring to the tolerance or context.
• If the method has limitations, mention them briefly and suggest how they might affect the result.