Indices, standard form, and surds

AQA GCSE Maths

8 ready-made resources for teaching Indices, standard form, and surds, written for AQA GCSE Maths. Slides, worksheets, mark schemes, homework, an assessment and retrieval quizzes, all with SEN-friendly scaffolding built in as standard. Below is what the topic requires and where students usually go wrong, free to read whether or not you sign up.

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What is in this bundle

Assessment mark schemeWord, editable
End-of-topic assessmentWord, editable
HomeworkWord, editable
Retrieval-practice quizzesWord, editable
SlidesPowerPoint, editable
Starter & exit ticketWord, editable
Tiered worksheetWord, editable
Worksheet mark schemeWord, editable

Foundation tier

Use the law of indices for multiplying powers of the same base (aᵐ × aⁿ = aᵐ⁺ⁿ).
Use the law of indices for dividing powers of the same base (aᵐ ÷ aⁿ = aᵐ⁻ⁿ).
Use the law of indices for raising a power to a power ((aᵐ)ⁿ = aᵐⁿ).
Know that any non-zero number raised to the power 0 equals 1.
Understand and use negative indices to represent reciprocals (a⁻ⁿ = 1 ÷ aⁿ).
Convert a number between ordinary form and standard form (a × 10ⁿ, where 1 ≤ a < 10).
Compare and order numbers given in standard form.
Multiply and divide numbers given in standard form.

Higher tier, in addition

Use fractional indices, including unit fractions (a^(1/n) = the nth root of a) and general fractional indices (a^(m/n) = (the nth root of a)ᵐ).
Add and subtract numbers given in standard form.
Solve problems set in real-life contexts (for example scientific measurements) that involve standard form.
Simplify a surd by extracting its largest square-number factor (for example write √50 as 5√2).
Multiply and divide surds, simplifying the result where possible.
Add and subtract surds where the surd part already matches.
Rationalise a denominator of the form 1 ÷ √a.

Where students go wrong

Adding or subtracting indices when the bases are different (the index laws only apply when the base is the same).
Confusing a⁰ = 1 with a × 0 = 0.
Treating a negative index as making the value negative, rather than as a reciprocal (a⁻ⁿ = 1 ÷ aⁿ, not −aⁿ).
Writing a standard form coefficient outside the range 1 ≤ a < 10 (for example 35 × 10⁴ instead of 3.5 × 10⁵).
Getting the sign of the power of 10 the wrong way round when converting numbers smaller than 1 to standard form.
Assuming √a + √b = √(a + b); surds can only be combined directly when the surd part already matches.
Leaving a surd partially simplified (for example stopping at √50 instead of continuing to 5√2).

How it gets asked in the exam

"Write ... as a single power of ...", "Write ... in standard form", "Work out ..., giving your answer in standard form", "Simplify √...", "Rationalise the denominator of ...", "Without using a calculator, work out ...".

Key vocabulary

Index, power, exponent, base, index law, reciprocal, standard form, ordinary form, coefficient, surd, rational number, irrational number, rationalise, simplify.

Assumed prior knowledge

Multiplication and division facts, and secure place value with large and small numbers.
Squares, cubes, and the square/cube roots of common numbers.
Powers as repeated multiplication.

How Speca scaffolds this topic

Provide a colour-coded "index laws card" with three boxes, multiply powers → add the indices; divide powers → subtract the indices; power of a power → multiply the indices, so students pattern-match the operation shown on the powers to the operation required on the indices, mirroring the transformation "ID card" scaffold used in the transformations and constructions topic.
For standard form conversions, use a consistent visual of the decimal point physically sliding left or right across the digits, with each place counted aloud and matched directly to the power of 10, so the exponent's value is seen rather than only memorised as a rule.
For surds, teach a fixed "factor pair finder" routine: always check a short reference list of square numbers (4, 9, 16, 25, 36, 49...) for the largest one that divides the number under the root before simplifying, preventing the common error of stopping at a partially-simplified surd.

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