A numerical answer can look reasonable and still fail the relationship stated in the question. A useful check returns to that original relationship instead of repeating the same calculation. Apply the discount to the proposed original price, substitute a value into the starting equation or test whether a boundary belongs to an inequality. Each check targets a different kind of error.
This practice guide supports the Shahdadpur O and A Level family plan and the broader O Level Maths checking guide. All exercises below are original teaching examples, not official examination items. Use the methods and answer forms required by the actual course task. A check supports reasoning; it does not replace requested working.
Identify what would make a proposed answer valid
Before checking, state what the answer must satisfy. In an equation, both sides must agree when the candidate is substituted, and the original expression must be defined. In a percentage problem, the proposed amount must reproduce the supplied change. In an inequality, the result describes a set of values rather than one isolated number.
This first sentence helps choose the check. Repeating an arithmetic operation may be useful for a suspected key-entry slip, but it may not detect the wrong percentage base or a lost domain restriction. Select a relationship that tests the concern directly. A check is strongest when the student can explain what a successful result would establish and what it would leave unresolved.
Keep the original task visible while doing this. A student may correctly check a value against an altered equation that already contains the error. Return to the actual starting statement wherever possible. That makes the check less dependent on the same rearrangement or interpretation used in the first attempt.
Reverse a percentage change by applying it forward
Take an invented price of 80 currency units after a 20 per cent discount. The original price is 100 units because 80 is 80 per cent of 100. To check it, calculate 20 per cent of 100, obtaining 20, and subtract that discount from 100. The result is the supplied 80. The proposed original amount reproduces the stated process.
Now inspect the common incorrect proposal of 96, obtained by adding 20 per cent of 80. A 20 per cent discount on 96 is 19.2, leaving 76.8. It does not reproduce 80. The check identifies the wrong base: the discount percentage applies to the original price, not the already discounted amount.
For a fresh example, suppose a fictional amount becomes 90 after a 25 per cent reduction. The original is 120 because 75 per cent of 120 is 90. Adding 25 per cent of 90 instead gives 112.5, and reducing that by 25 per cent gives 84.375. The mismatch exposes the same misconception with new values rather than merely repeating the first answer.
Substitute into the original fractional equation
Consider the equation (x + 1)/(x − 1) = 3. The original expression requires x not equal to 1 because division by zero is undefined. Multiplying by x − 1 within that allowed domain gives x + 1 = 3x − 3. Rearranging yields 4 = 2x, so x = 2.
Check x = 2 in the original equation: (2 + 1)/(2 − 1) = 3/1 = 3. The candidate is allowed and satisfies the equality. Record both checks. A value that makes a denominator zero cannot be accepted merely because it appears to work in an equation obtained after multiplying away that denominator.
The restriction should remain visible throughout the solution. Do not wait for an error message to decide what the expression means. The changed-condition guide offers older students related examples involving nonzero conditions. For the present exercise, focus on the original denominator and the candidate's validity.
Check candidates introduced by squaring
Use another original equation: √(x + 6) = x, over the real numbers, with the square root interpreted as its nonnegative principal value. A solution must have x at least zero because the left side is nonnegative. Squaring gives x + 6 = x², or x² − x − 6 = 0. Factoring gives candidates x = 3 and x = −2.
Substitute both into the original equation. For x = 3, √9 = 3, so the equality holds. For x = −2, the left side is √4 = 2, which is not −2. The negative candidate is extraneous for the original equation even though it satisfies the squared equation. The original equation has the single real solution x = 3.
Explain why the check matters. Squaring can make two unequal opposite values become equal, so an equation after squaring may admit extra candidates. Do not reject a value merely because it is negative in every problem; here the original nonnegative square root and equality justify the rejection. The condition must come from the task, not a habit of discarding negative answers.
Check an inequality as a set of values
Consider 2 − 3x > 11. Subtracting 2 gives −3x > 9. Dividing by −3 reverses the inequality direction, producing x < −3. A student who keeps the greater-than symbol would report the wrong side of the boundary. Testing values can expose the error while the algebra explains the full solution set.
Try x = −4, which belongs to x < −3. The original left side becomes 14, and 14 > 11 is true. Try x = −2, which lies on the other side. The left side becomes 8, and 8 > 11 is false. At x = −3, the left side is 11, so the strict inequality is false. The boundary is excluded.
These tests support the direction and boundary interpretation. A few tested values alone do not prove the full interval; the valid algebra establishes it. If the question asks for a number-line representation, use the appropriate open boundary and direction. If it asks for integer solutions in a restricted range, apply that additional domain rather than reporting every real value in the interval.
Use units and context to inspect the requested quantity
Suppose an original rectangle exercise gives length 8 centimetres and width 3 centimetres. Its area is 24 square centimetres, while its perimeter is 22 centimetres. Both calculations use the same supplied dimensions, but they answer different questions. Reporting 24 with a length unit would also signal a mismatch between the operation and the quantity.
Read the final instruction before accepting the number. A student may calculate a useful intermediate quantity and then stop. A check should identify what that intermediate value represents and whether another step is needed. Correct arithmetic does not compensate for answering area when the task asks for perimeter.
Unit analysis is useful where quantities have units, but it is not a complete proof. Two wrong expressions can share the same units. Use a relevant estimate or original relationship as another check when needed. The calculator-check guide gives older students further examples of choosing a check that addresses the actual suspected error.
Correct the first invalid step rather than only the result
When a check fails, inspect the working from the beginning and find the first unsupported transition. In the inequality example, that may be the unchanged symbol after dividing by a negative quantity. In reverse percentages, the error occurs earlier, when identifying the base. Changing only the final answer hides the cause and makes the same problem more likely to remain unnoticed in fresh work.
Keep the original attempt and add a concise correction. Then choose a new task requiring the same decision. If the student can now explain the base, denominator restriction or inequality reversal without a hint, that is useful evidence of the repair. If help is still needed, record what it supplied and discuss that specific connection with the teacher.
The Shahdadpur answer-key routine explains how a family can preserve this evidence during review. The specific teacher-question guide helps turn a failed check into an actionable question. Bring the original equation or wording, not just the final numbers, so the relevant conditions remain visible.
Keep checking purposeful and proportionate
For a short session, choose one percentage problem, one equation and one inequality only if the student has studied the necessary content. Ask which check fits each task before revealing the model. The aim is to practise selecting a check, not to make every answer carry an identical long checklist regardless of its structure.
Use the Pakistan O Level revision plan to schedule a later independent revisit. Confirm subject support and any work-review arrangement through the admissions FAQ. This guide does not establish a local class, marking entitlement or examination service in Shahdadpur. Use the student's actual work to decide which relationship needs the next explanation.
Frequently asked questions
Is repeating my calculation enough to check it?
It can detect some arithmetic or entry errors, but it may repeat the same wrong setup. Return to the original relationship through substitution, a forward percentage check or another relevant test. Explain what the check is intended to detect.
Why can squaring create an extra candidate?
Squaring can remove a sign distinction between opposite values. A value satisfying the squared equation may fail the original equation. Substitute each candidate into the original expression and apply its stated domain and sign conditions.
Can a denominator restriction be ignored after rearranging?
No. The original expression must be defined. Multiplying away a denominator does not make a previously excluded input valid for the original equation. Retain the restriction and check any proposed solution against it.
Do test values prove an entire inequality solution?
Not by themselves in a general real-number domain. They can expose a wrong direction or boundary interpretation. Valid algebra or another appropriate general argument establishes the full set, with test values used as supporting checks.
What should I record when a check fails?
Keep the original attempt, identify the first invalid step and explain the correction briefly. Then attempt a fresh related task without the model. Record meaningful hints so that the next support decision reflects what the student can do independently.
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