How to calculate ESUA, units, standard form, sig figs and rearranging

SPEC 8463 • EXAM FOUNDATIONS

How to Calculate

ESUA — the four-step calculation framework that mirrors AQA mark schemes exactly. Plus units, prefixes, standard form, sig figs and rearranging — the four supporting skills that protect every calculation mark you earn.

Why ESUA — Connecting To The Mark Scheme

In lesson 0.01 you learned that calculations are marked M1 → M1 → A1. ESUA is the answer-side version of that. Every calculation question, every paper — the same four-step rhythm wins the marks.

How ESUA Maps To The Mark Scheme

ESUA StepWhat you writeMark earnedE — EquationSymbols (e.g. F = m a)M1S — SubstituteNumbers in SI unitsM1U + A — Unit & AnswerNumber + correct unitA1 (single mark, awarded together)

Important

ESUA has four steps but earns three marks on a typical 3-mark calculation, not four. U and A combine into a single A1: a number on its own scores 0; a unit on its own scores 0; together they score 1. On longer (4–5 mark) calculations — especially those that say "Give the unit" with a separate unit line — the unit can be its own A1 mark. The discipline is the same either way: always state both.

Key Definitions

Four terms that appear directly on mark schemes. The exact wording matters because answers in non-SI units, or with too many digits, can lose the A1 even when the number is right.

SI Units

The standard system of units used in physics: m (length), kg (mass), s (time), N (force), J (energy), W (power), A (current), V (potential difference), Pa (pressure), Hz (frequency), °C (temperature, used throughout the GCSE Physics spec).

Standard Form

A number written as A × 10ⁿ, where 1 ≤ A < 10. Used for very large or very small numbers (e.g. 5.6 × 10⁴).

Significant Figures

Digits that carry meaningful precision. Final answers are quoted to 2 or 3 sig figs — matched to the precision of the data given.

Prefix

A letter placed before a unit to multiply it by a power of 10 — e.g. "k" in kJ means × 10³. Full table on slide 8.

The ESUA Chain

The four steps, in order. Build this rhythm into every calculation in your homework and notes from today, and the exam version becomes automatic.

Equation

Write the formula in symbols.

Substitute

Convert units first, then plug in numbers.

A1*

Unit

State the SI unit of the answer.

A1*

Answer

Quote to 2–3 sig figs; standard form if needed.

* U and A are awarded together as one A1 — number + unit = the single answer mark.

The Order Matters

Equation FIRST — because if you start with numbers, you can't show the examiner what relationship you're using. The mark scheme is reading line-by-line: line 1 = M1, line 2 = M1, line 3 = A1.

Step 1 — Equation (M1)

Write the correct equation in symbols, even when it appears on the equation sheet. This is the easiest mark in the whole calculation — and the one most often skipped.

F = m a | V = I R | E_k = ½ m v² | ρ = m / V

What Earns M1

The equation in symbols (the formula form), OR

The equation rearranged for the unknown (e.g. a = F / m), OR

The equation in words if you can't remember the symbols (e.g. "force = mass × acceleration") — usually still credited

Common Failure

Skipping straight to the substitution: writing F = 2.0 × 9.8 with no equation line above. This loses M1 — the examiner can't tell which formula you're applying. Equation first, every time.

Step 2 — Substitute (M1)

Replace the symbols with their numerical values in SI units. This is the highest-value step in the calculation because any unit error now corrupts everything downstream — substitution + answer + unit. Lose this M1 and the wrong number flows into A1 too.

The Two-Part Substitution

Convert any non-SI values to SI units (cm → m, g → kg, min → s, kJ → J, etc.)

Substitute the converted values into the equation

Convert before the substitution line, not after. The substitution line should already contain SI numbers.

The Single Biggest Mark-Loser

Examiner reports return to this every year: students substitute non-SI values directly (e.g. 250 grams treated as 250, or 80 cm treated as 80) and produce answers that are orders of magnitude wrong. The substitution step is the place to convert — never at the end as a "fix".

For Multi-Step / Chained Calculations

High-demand questions often chain 2–3 equations (e.g., F = ma → v² − u² = 2as, or Power → Energy → SHC). The rule: carry full calculator precision through every intermediate step, then round only the final answer. Rounding at each step compounds errors — and can cost you A1 even when the method is right. (More on chained calcs in Module 1.)

Unit Conversions — The Table To Memorise

These conversions appear in nearly every calculation question. Memorise them now and you eliminate the largest single source of lost marks across both papers.

FromToOperation

cmm÷ 100

mmm÷ 1000

kmm× 1000

gkg÷ 1000

mgkg÷ 1 000 000 (× 10⁻⁶)

kJJ× 1000

MJJ× 1 000 000 (× 10⁶)

kWW× 1000

mWW÷ 1000

minutess× 60

hourss× 3600

kPaPa× 1000

MPaPa× 1 000 000 (× 10⁶)

km/hm/s÷ 3.6

Memory Trick

"k means kilo means × 1000 going up; ÷ 1000 going down." Same logic for M (mega = × 10⁶), G (giga = × 10⁹), m (milli = × 10⁻³), μ (micro = × 10⁻⁶).

SI Prefixes — Powers Of 10 In One Chart

Every prefix is just shorthand for a power of 10. Understand the pattern and you don't need to memorise individual conversions — you derive them.

Worked Examples

320 MHz = 320 × 10⁶ Hz = 3.2 × 10⁸ Hz

50 nm = 50 × 10⁻⁹ m = 5.0 × 10⁻⁸ m

2.5 kW × 4.0 minutes = 2500 W × 240 s = 600 000 J = 6.0 × 10⁵ J

Standard Form — The Rules

Standard form is required when numbers get very large (electricity, space) or very small (atoms, wavelengths). The rule is simple, but the arithmetic in the powers is where calculator-free students lose marks.

The Format

A × 10ⁿ where 1 ≤ A < 10

Examples: 56 000 = 5.6 × 10⁴ · 0.000 042 = 4.2 × 10⁻⁵ · 3.0 × 10⁸ (speed of light, m/s).

Multiplication

Multiply the front numbers · ADD the powers.

(2 × 10³) × (3 × 10⁴)

= 6 × 10⁷

Division

Divide the front numbers · SUBTRACT the powers.

(3 × 10⁸) ÷ (1.5 × 10⁵)

= 2 × 10³

Calculator Tip

On most exam calculators, the standard-form key is labelled EXP or ×10ⁿ. To enter 3 × 10⁸ press 3 EXP 8 — NOT 3 × 10 EXP 8 (that gives you 3 × 10⁹). One of the most common single-keystroke errors in the exam.

Step 3 + Step 4 — Unit & Answer (A1 together)

U and A combine into a single A1 mark. A number on its own scores 0; a unit on its own scores 0; the two together earn the answer mark. Both have to be right.

U — The Unit

Always state the SI unit of your answer. The most common slips:

m/s not "m per s" or omitted

J not "joules" written longhand (both are accepted but the symbol is safer)

N m for moments — a space between the symbols, not "Nm"

If your substitution used non-SI units, your derived unit will be non-SI too — convert FIRST

A — Significant Figures

Match the precision of the question's data:

Question gives values to 2 sig fig → answer to 2 sig fig

Question gives values to 3 sig fig → answer to 3 sig fig

Mixed precision → match the least precise input

Use standard form when the answer is very large (≥10⁴) or very small (≤10⁻³)

Don't Quote Spurious Precision

If the question gives m = 5.0 kg (2 sig fig), don't quote your answer to 5 decimal places — your calculator's precision exceeds the data's, so those extra digits are meaningless. 2.0 J, not 1.96400 J.

Rearranging — The Universal Method

You'll rearrange equations dozens of times across the course. The method never changes: do the same operation to both sides until your unknown is alone. Like a balance scale — whatever you do to one side, do to the other.

The Procedure (3 Steps)

Identify what's "stuck" to your unknown. Multiplication? Division? Squared?

Apply the inverse — multiplication ↔ division, square ↔ square root, addition ↔ subtraction.

Apply it to both sides simultaneously. If you do it to one side only, the equation is no longer true.

Rearranging — Two Worked Examples

The procedure is the same for every equation, but the inverse operation depends on what's stuck to your unknown. Two cases worth practising explicitly.

Case 1: Make c the subject of ΔE = m c Δθ.

Working Through The Steps

Start: ΔE = m c Δθ

c is multiplied by m and by Δθ → divide both sides by m and by Δθ.

÷ m Δθ both sides: ΔE / (m Δθ) = c

Flip: c = ΔE / (m Δθ)

Case 2 (HT): Make v the subject of E_k = ½ m v².

The Square Root Case

Start: E_k = ½ m v²

Two operations stuck to v: it's squared AND multiplied by ½m. Reverse them in order — multiplication first, then square.

× 2 / m both sides: 2 E_k / m = v²

√ both sides: v = √(2 E_k / m)

Best Practice

Rearrange in symbols first, THEN substitute numbers. Trying to rearrange while the numbers are in is a recipe for errors — and many mark schemes specifically award credit for the rearranged symbolic form before any numbers go in.

ESUA — Full Worked Example

A classic three-mark calculation that tests every part of ESUA: equation recall, two unit conversions, the substitution, and a sig-fig-aware answer.

A 250 g mass falls through a height of 80 cm. Calculate the gravitational potential energy lost. Take g = 9.8 N/kg. 3 marks

The Four Steps In Action

EEquation:Eₚ = m g h(M1)SSubstitute — first convert: m = 250 g = 0.250 kg · h = 80 cm = 0.80 m. Then plug in: Eₚ = 0.250 × 9.8 × 0.80(M1)UUnit: joules (J)AAnswer:Eₚ = 1.96 J ≈ 2.0 J (2 sig fig — the mass and height were both given to 2 sig fig). (A1)

Three marks total: M1 (equation) + M1 (substitution with conversion) + A1 (answer + unit together).

Examiner Alert — Convert At The Substitution Step

The single biggest source of lost calculation marks across the qualification. Examiner reports identify it again and again. The rule is simple — but the temptation to "convert later" is real.

Many candidates substituted values directly without first converting to SI units. A frequent pattern was substituting masses in grams or distances in centimetres into equations expecting kilograms and metres, producing answers wrong by a factor of 10² or 10³. — pattern reported across multiple AQA examiner reports, including Jun 2024

The Discipline

Read the question. Note any non-SI units (cm, mm, g, mg, kJ, kW, min, h, kPa).

Before writing the substitution line, convert each value and write the SI equivalent.

Substitute the converted values into the equation.

Never multiply or divide by a power of 10 at the end of your working to "fix" a wrong-magnitude answer — that's where errors compound, not where they get solved.

The Show-Working Principle & ECF

AQA's marking guidance has two clauses every Physics student should know — they explain why working matters, and the second one is the part most students never hear.

Marks should be awarded for each stage of the calculation completed correctly... Full marks can, however, be given for a correct numerical answer, without any working shown. — AQA mark scheme principles, §3.3 (every Physics 8463 mark scheme)

Any error in the answers to a structured question should be penalised once only... Allowances for errors carried forward (ECF) are most likely to be restricted to calculation questions. — AQA mark scheme principles, §3.5

What This Means In Practice

Correct answer alone CAN earn full marks. AQA's principle allows it.

BUT a wrong answer with no working = 0 marks. No method marks are recoverable. One slip = lose everything.

ECF (Errors Carried Forward): on chained / multi-part questions, if you carry a wrong value through correctly, you still earn the downstream marks. This only works if your working is visible.

The Strategic Conclusion

Showing working has no downside. If your answer is right, you get full marks (with or without working). If your answer is wrong, working salvages method marks and protects ECF on every later part. Working is insurance against your own slip-ups — and it's free.

Weak vs Strong — When Conversion Fails

Same calculation, same equation. The student who didn't convert before substituting still showed working — so they earn the equation mark. But the wrong substitution costs M1, and the wrong answer costs A1. Working preserves what it can.

A 250 g mass falls 80 cm. Calculate Eₚ. (g = 9.8 N/kg)3 marks

✗ ~1 / 3 marks

Eₚ = m g h ✓ M1

Eₚ = 250 × 9.8 × 80 ✗ no conversion

Eₚ = 196 000 ✗ wrong by ×10⁵

Eₚ = 196 000 J

M1 earned for the equation. Substitution M1 lost — values are not in SI. A1 lost — answer is roughly 100 000 times too big. Working at least preserved the equation mark.

✓ 3 / 3 marks

Eₚ = m g h ✓ M1

m = 0.250 kg · h = 0.80 m

Eₚ = 0.250 × 9.8 × 0.80 ✓ M1

Eₚ = 1.96 J ≈ 2.0 J ✓ A1

Conversion BEFORE substitution. SI numbers in the substitution line. Answer to 2 sig fig with correct unit.

The Habit That Saves These Marks

Whenever you read a number with a non-SI unit, immediately write its SI equivalent on your scratch space — before you start the calculation. Two seconds of conversion now prevents two marks lost later.

Practice Question

A kettle's heating element is rated at 2.5 kW. The kettle takes 4.0 minutes to boil a quantity of water.

Calculate the energy transferred to the water during this time. Give your answer in joules and in standard form. 3 marks

E

Equation:E = P t. This earns M1 even though E = P t appears on the 2025–2027 equation sheet — write the formula first.

S

Substitute — convert first: P = 2.5 kW = 2500 W · t = 4.0 min = 4.0 × 60 = 240 s. Then: E = 2500 × 240. This earns the second M1 — both conversions correct, SI values substituted.

U + A

Unit and Answer:E = 600 000 J = 6.0 × 10⁵ J (2 sig fig). This earns A1 — number in standard form with correct unit.

⚠️

Mark loss check: writing just "600 000 J" is acceptable but the question asks for standard form. Writing "6.0 × 10⁵" without the J loses A1. Substituting 2.5 × 4.0 = 10 J (no conversion of either) gets you a wrong answer 60 000× too small.

Common Errors — Quick Recap

The patterns that cost marks across both papers. If you build the habit of checking each one before moving on from a calculation, you protect 4–6 marks per paper.

✗ Common Error✓ What To Write Instead

Skipping the equation lineAlways write the equation in symbols FIRST — even when on the sheet

Treating 250 g as 250 in F = maConvert: 250 g = 0.250 kg, THEN substitute

Substituting then "fixing" with × 1000 at the endConvert at the substitution line, never as a post-hoc fix

Final answer with no unitAlways state the SI unit — missing unit can lose A1

Quoting 1.96400 J when data is 2 sig figMatch the precision of the data (e.g. 2.0 J)

Calculator entry: 3 × 10 EXP 8 (gives 3 × 10⁹)Use the EXP key directly: 3 EXP 8 = 3 × 10⁸

Rearranging while numbers are in the equationRearrange in symbols, THEN substitute

Summary

The ESUA Framework

E — Equation in symbols (M1)

S — Substitute SI values, converted FIRST (M1)

U + A — Unit and Answer together = single A1 (typical 3-mark calc)

Three steps written, three marks earned. Skip a step, lose a mark.

The Five Supporting Skills

Unit conversions — cm/mm/g/min/kJ/kW/kPa to SI (table on slide 7)

SI prefixes — G(×10⁹), M(×10⁶), k(×10³), m(×10⁻³), μ(×10⁻⁶), n(×10⁻⁹)

Standard form — A × 10ⁿ with 1 ≤ A < 10; multiply = add powers; divide = subtract powers

Sig figs — match the precision of the question's data (usually 2 or 3)

Rearranging — same operation to both sides; in symbols first, then substitute. Square root for v in E_k = ½mv².

The Marking Principles That Matter

Correct answer alone can earn full marks — but if it's wrong, no working = 0 marks

ECF (Errors Carried Forward): a wrong value carried correctly through later steps still earns those marks — only works if working is shown

Working is insurance , not a marker requirement: zero downside, large upside

The Three Mark-Saving Habits

Write the equation FIRST, even when it's on the sheet

Convert to SI BEFORE the substitution line — never at the end

Always write the unit alongside the final number — A1 needs both

Checkpoint — You Can Now

Apply ESUA to any GCSE Physics calculation and map each step to its M1/M1/A1 mark

Convert routinely between SI and non-SI units (cm/mm/g/kJ/kW/min/h/kPa)

Use SI prefixes and standard form, including multiplication and division

Rearrange any spec equation, including the square-root case (E_k → v)

Explain AQA's marking principle and ECF — and why showing working is defensive insurance