lesson
One example can disprove an always claim
1 · Learn
Testing several examples can suggest a pattern, but it does not prove a statement for every permitted value. A counterexample disproves an always claim. An algebraic argument can establish why a statement holds under stated conditions.
For integer n, the sum of consecutive integers n and n + 1 is 2n + 1, which is odd. This explains all integer cases rather than only the first few.
2 · Worked example
The claim all prime numbers are odd is false because 2 is prime and even. This single valid counterexample is enough.
3 · Your turn
Explain algebraically why the sum of two even integers is even.
Check your answer
Write them as 2a and 2b for integers a,b. Their sum is 2(a + b), a multiple of two.
4 · Apply your learning
Distinguish a convincing example from a general proof and identify the domain each argument uses.
Adult guidance and safety
Require reasoning and a check. Provide accurate graphs, construction instruments and varied practice. Annual KS3 allocation is proposed; do not treat these short models as a complete assessed programme.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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