Pearson Edexcel · A-Level · Mathematics

    Course version: 9MA0 · Version from Mar 2026

    Mathematical problem solving: Evaluate, including by making reasoned…

    Full specification wording

    Overarching theme 2: Mathematical problem solving: Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions, including those obtained using numerical methods.

    Evaluating a solution means judging how trustworthy it is, not merely calculating it.

    Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .

    • Point 5 of 7
    • 3 key terms

    Study notes

    From the checked teaching notes for this specification point.

    The idea

    Evaluating a solution means judging how trustworthy it is, not merely calculating it. For a numerical root, you can bound the error: after n bisection steps on [a, b], the absolute error is at most (b − a) / 2ⁿ. For Newton–Raphson, convergence may be rapid near a simple root but can fail if the derivative is small or the starting value is poor. Reasoned estimates include checking the sign and size of the answer, comparing with a graph, or using a simpler model. Limitations may arise from rounding, iteration count, model assumptions or sensitivity to initial data. You should state what the solution does and does not tell you, and whether the accuracy is sufficient for the context.

    What earns the marks

    • States a numerical bound or estimate for the error in a solution, such as an interval width or a function-value tolerance.
    • Compares an approximate solution with an independent check, for example a graph, a sign change or a simpler estimate.
    • Identifies limitations of a method, including slow convergence, failure near a zero derivative or sensitivity to the starting value.
    • Distinguishes between error from the numerical method and error from rounding or from the mathematical model itself.
    • Judges whether the achieved accuracy is adequate for the stated purpose and communicates that judgement clearly.

    Common mistakes and how to put them right

    • Claiming an answer is exact when it is numerical.

      Put it right: Quote the tolerance and describe the result as accurate to that tolerance.

    • Ignoring rounding error accumulated over many iterations.

      Put it right: Consider how rounding affects the final accuracy.

    • Assuming a method always converges.

      Put it right: Check conditions such as a sign change or a non-zero derivative before relying on the result.

    Exam tips

    • Give a numerical bound on the error rather than a vague statement that the answer is close.
    • Use a graph or a sign check to support your evaluation of accuracy.
    • Comment explicitly on whether the accuracy is sufficient for the problem context.

    Try the lesson

    The start of the Overarching theme 2: Mathematical problem solving: Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions, including those obtained using numerical methods. lesson, made from checked teaching notes for this specification point.

    Have a go first

    A numerical method gives the root of an equation as 3.1416, correct to 4 decimal places. Which statement best describes the accuracy of this value?

    Listen

    Evaluating a solution means judging how trustworthy it is, not merely calculating it. For a numerical root, you can bound the error: after n bisection steps on [a, b], the absolute error is at most (b − a) / 2ⁿ. For Newton–Raphson, convergence may be rapid near a simple root but can fail if the derivative is small or the starting value is poor. Reasoned estimates include checking the sign and size of the answer, comparing with a graph, or using a simpler model. Limitations may arise from rounding, iteration count, model assumptions or sensitivity to initial data. You should state what the solution does and does not tell you, and whether the accuracy is sufficient for the context. A strong answer earns marks for these points. States a numerical bound or estimate for the error in a solution, such as an interval width or a function-value tolerance. Compares an approximate solution with an independent check, for example a graph, a sign change or a simpler estimate. Identifies limitations of a method, including slow convergence, failure near a zero derivative or sensitivity to the starting value. Distinguishes between error from the numerical method and error from rounding or from the mathematical model itself. A common slip: Claiming an answer is exact when it is numerical. Instead, quote the tolerance and describe the result as accurate to that tolerance. Exam tip: Give a numerical bound on the error rather than a vague statement that the answer is close.

    See it

    A top-down flowchart starting from a single node labelled Numerical solution, which branches into three nodes: Bound the error, Independent check and Identify limitations. Bound the error leads to a node stating the bisection bound of (b − a) / 2ⁿ. Independent check leads to a node listing a graph, sign change or simpler estimate. Identify limitations leads to two nodes: one on slow convergence, zero derivative or poor starting value, and one on rounding, iteration count, model assumptions or sensitivity to initial data. All four of these end nodes feed into a final node: Judge whether accuracy is sufficient for the context.
    This shows how a numerical solution is evaluated by bounding the error, checking independently, identifying limitations and judging whether the accuracy is sufficient.

    Key words

    Say what each word means before you turn the card.

    Error bounds

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    Reasoned estimation

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    Limitations of numerical methods

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    See every definition
    Error bounds
    A numerical limit on how far a solution may be from the true value, such as an interval width or a function-value tolerance.
    Reasoned estimation
    Judging accuracy by checking the sign and size of the answer, comparing with a graph, or using a simpler model.
    Limitations of numerical methods
    Drawbacks such as slow convergence, failure near a zero derivative or sensitivity to the starting value.

    Label the diagram

    Some boxes have lost their names. Pick the right name for each number.

    The diagram again, with 3 boxes numbered instead of named. A top-down flowchart starting from a single node labelled Numerical solution, which branches into three nodes: Bound the error, Independent check and Identify limitations. Bound the error leads to a node stating the bisection bound of (b − a) / 2ⁿ. Independent check leads to a node listing a graph, sign change or simpler estimate. Identify limitations leads to two nodes: one on slow convergence, zero derivative or poor starting value, and one on rounding, iteration count, model assumptions or sensitivity to initial data. All four of these end nodes feed into a final node: Judge whether accuracy is sufficient for the context.

    • 1Box 1

    • 2Box 2

    • 3Box 3

    0 of 3 labelled

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