lesson
Look beyond the first difference
1 · Learn
A sequence with constant second differences can be modelled by a quadratic rule. For n², the terms 1, 4, 9, 16 have first differences 3, 5, 7 and constant second difference 2.
A finite list does not uniquely prove an infinite rule. State the proposed rule and test it against every supplied term rather than claiming it is the only possible continuation.
2 · Worked example
The terms 3,6,11,18 match n² + 2 for n starting at 1. Substitution checks all four terms.
3 · Your turn
Using the stated rule n² + 2, find the fifth and tenth terms.
Check your answer
27 and 102.
4 · Apply your learning
Compare n², 2n² and n² + 3n using difference tables and explain what the tables suggest.
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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