lesson
Repeated changes multiply
1 · Learn
Increasing an amount by 5% multiplies it by 1.05. Repeating the change multiplies the new amount again, so two changes use 1.05². The second increase is not based on the original amount.
A rise and fall of the same percentage do not cancel because their bases differ. State how many changes occur and when rounding is applied.
2 · Worked example
£200 increased by 10% becomes £220. A subsequent 10% reduction gives £198, not £200.
3 · Your turn
A population model starts at 800 and grows by 5% twice. Find the modelled final value.
Check your answer
800 × 1.05² = 882.
4 · Apply your learning
Compare repeated multiplication with adding two percentages of the original, explaining the difference.
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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