O-Level Physics Paper 3 Practical Maths Toolkit

> **TL;DR**\
> SEAB’s Mathematical Requirements for Physics (6091) expect Paper 3 candidates to handle standard form arithmetic, rearrange formulae, apply trig, and extract gradients from straight-line plots with ease ([SEAB 2026 Physics syllabus](https://www.seab.gov.sg/files/O%20Lvl%20Syllabus%20Sch%20Cddts/2026/6091_y26_sy.pdf)).\
> Build repeatable routines: sanity-check significant figures, change subject with SI units in mind, map curved relationships into linear form, and quote gradients/intercepts with proper units.\
> Use the drills below to keep calculator workflows sharp—no more stalled planning responses when a reciprocal plot or sin⁻¹ step appears mid-exam.

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## 1 | Arithmetic moves Paper 3 markers assume you can do

- **Standard form & powers:** Practise multiplying/dividing numbers such as \\( (3.2 \times 10^\{-3}) \times (4.5 \times 10^{2}) \\) without losing significant figures ([SEAB 2026 Physics syllabus, Mathematical Requirements](https://www.seab.gov.sg/files/O%20Lvl%20Syllabus%20Sch%20Cddts/2026/6091_y26_sy.pdf)).\\
- **Averages & reciprocals:** When you take repeated readings, compute the mean and reciprocal quickly (e.g. average \\( 1 / t \\) for pendulum timing).\\
- **Root and trig keys:** Confirm your calculator (from the approved list) can toggle between degree mode, square roots, and inverse trig functions; reset to degrees before the exam session.\\
- **Approximation sense-check:** Before committing, round values to 1 s.f. to anticipate expected magnitudes—if your calculated density is \\( 7.6 , \text\{g cm^\{-3}} \\), an estimated \\( 8 , \text\{g cm^\{-3}} \\) confirms the order of magnitude is reasonable (same source).

**Quick drill:** In 60 seconds, convert three raw numbers into standard form, compute their product/quotient, and round to the correct significant figures. Repeat until error-free.

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## 2 | Algebra that powers Planning and ACE responses

### 2.1 | Changing subject with units intact

| **Scenario**       | **Original relation**                                    | **Rearranged for**                                        | **Reminder**                                           |
| ------------------ | -------------------------------------------------------- | --------------------------------------------------------- | ------------------------------------------------------ |
| Density comments   | \\( \rho = \dfrac\{m}\{V} \\)                            | \\( m = \rho V \\)                                        | Keep SI units consistent (kg and m³).                  |
| Resistivity plan   | \\( R = \rho \dfrac\{L}\{A} \\)                          | \\( \rho = R \dfrac\{A}\{L} \\)                           | Convert area from mm² to m² before substitution.       |
| Lens linearisation | \\( \dfrac\{1}\{f} = \dfrac\{1}\{u} + \dfrac\{1}\{v} \\) | \\( \dfrac\{1}\{v} = -\dfrac\{1}\{u} + \dfrac\{1}\{f} \\) | Ready for straight-line plotting (\\( y = mx + c \\)). |

### 2.2 | Proportion arguments in planning

- Use direct proportion to justify graph choices, e.g. “If \\( V \propto I \\), plotting \\( V \\) against \\( I \\) should yield a straight line through the origin.”\\
- Invoke inverse proportionality when designing tables—if \\( T \propto 1 / \sqrt\{L} \\), include a processed column for \\( 1 / \sqrt\{L} \\).

### 2.3 | Algebra drill

1. Rearrange three formulae chosen from your current topic list, writing the SI unit check beside each.\\
2. Explain in one sentence how the rearranged form feeds your graph or calculation.\\
3. Time the exercise—keep it under 3 minutes.

All expectations above are explicitly called out under the Algebra bullet points in the Mathematical Requirements ([SEAB 2026 Physics syllabus](https://www.seab.gov.sg/files/O%20Lvl%20Syllabus%20Sch%20Cddts/2026/6091_y26_sy.pdf)).

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## 3 | Geometry & trig for mechanics, waves, and optics

| **Use case**                | **Math move**                                            | **Toolkit reminder**                                                                 |
| --------------------------- | -------------------------------------------------------- | ------------------------------------------------------------------------------------ |
| Resolved forces in planning | \\( \sum F = 0 \\) with \\( F \cos \theta \\) components | Draw a quick vector triangle; ensure calculator is in degrees; quote angles to 1°.   |
| Moments questions           | Lever arm \\( = r \sin \theta \\)                        | Convert cm to m before calculating moment in N m.                                    |
| Ripple tank or Snell’s law  | \\( \sin i / \sin r = n \\)                              | Record angles with protractor precision (1°) and propagate into \\( \sin \\) values. |
| Pendulum swing geometry     | Arc length \\( s = r \theta \\) (radians)                | Remember \\( \theta = s / r \\) when estimating small-angle displacement.            |

The syllabus requires familiarity with trig functions, Pythagoras’ theorem, and basic area/volume formulae, along with the ability to use a protractor and other instruments ([SEAB 2026 Physics syllabus, Mathematical Requirements](https://www.seab.gov.sg/files/O%20Lvl%20Syllabus%20Sch%20Cddts/2026/6091_y26_sy.pdf)).

**Trig sprint:** Given \\( i = 35^\circ \\) and \\( r = 21^\circ \\), compute \\( n \\) and its percentage uncertainty assuming \\( \pm 1^\circ \\) resolution. Repeat with downhill slope angles for mechanics drills.

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## 4 | Graph intelligence: from curve to straight line

- **Select variables wisely:** The requirements emphasise translating between algebraic and graphical forms and rearranging into \\( y = mx + c \\) ([SEAB 2026 Physics syllabus](https://www.seab.gov.sg/files/O%20Lvl%20Syllabus%20Sch%20Cddts/2026/6091_y26_sy.pdf)).\\
- **Create processed columns:** For a cooling curve using Newton’s law, add \\( \ln(\Delta T) \\) to linearise \\( \ln(\Delta T) = \ln(\Delta T_0) - kt \\).\\
- **Gradient meaning:** State what the gradient and intercept represent before plotting; e.g. gradient of \\( V \\) vs \\( I \\) equals resistance, intercept indicates internal EMF.\\
- **Tangent skills:** Practice drawing tangents for non-linear graphs (e.g. displacement–time of non-uniform motion) to estimate instantaneous velocity, as the syllabus expects tangent gradient familiarity.

**Graph drill:** Take one non-linear relation each week and work out how to linearise it, then sketch axis labels and expected gradient/intercept meaning before touching data.

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## 5 | Calculator hygiene and exam-day checklist

1. **Mode check:** Set to degrees; disable any lingering statistical modes.\\
2. **Reset memory:** Clear stored variables that might interfere with calculations.\\
3. **Shortcut practice:** Use `Ans` and memory recall to speed up repeated multiplications (e.g. \\( IVt \\) when stepping through heat-capacity data).\\
4. **Significant-figure discipline:** After each output, glance at the raw inputs to ensure you haven’t over-reported digits.\\
5. **Battery check:** Replace if the contrast or response slows—SEAB allows only approved calculators, and last-minute battery swaps remove one worry.

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## 6 | Weekly maths maintenance plan (30 minutes)

| **Day**   | **Focus**               | **Suggested drill**                                       | **Outcome**                           |
| --------- | ----------------------- | --------------------------------------------------------- | ------------------------------------- |
| Monday    | Arithmetic & sig figs   | 10 quick conversions + product/quotients in standard form | No hesitation during data processing. |
| Wednesday | Algebra & rearrangement | Change subject for three formulas, write unit check       | Faster planning responses.            |
| Friday    | Graph & gradient        | Linearise one relation, plot mini graph, compute gradient | Smoother PDO/ACE commentary.          |
| Weekend   | Mixed paper snippet     | Attempt a past Paper 3 maths-only segment                 | Consolidate in timed conditions.      |

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## 7 | Ready-to-print maths support sheet

```text
Paper 3 Maths Quick Sheet
-------------------------
1. Standard form rules (10^a × 10^b = 10^(a+b))
2. Rearrangement reminders (R = ρL/A → ρ = RA/L)
3. Trig flash (sin, cos, tan; inverse functions; degree mode)
4. Linearisation templates (e.g., ln y = ln a + bx)
5. Gradient checklist (triangle spans ≥ 1/2 line, units quoted)
6. Sig fig workflow (match least precise input)
7. Calculator mode + battery check
```

Keep a laminated copy in your revision folder and annotate it after each mock to capture personal pitfalls (e.g. “Remember to convert cm² to m² before plugging into resistance formula”).

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## Sources

1. [SEAB, GCE O-Level Physics (6091) Syllabus 2026 — Mathematical Requirements](https://www.seab.gov.sg/files/O%20Lvl%20Syllabus%20Sch%20Cddts/2026/6091_y26_sy.pdf)