The shape of the lesson
By the end of the lesson, learners can
- Weigh the units, not the numbers: Reduce each side of an equation to SI base units and put them on a balance.
- The base, the derived, the prefix: Five base quantities carry almost all of AS mechanics and electricity.
- Estimates the examiner expects.
- Base units: kg, m, s, A, K. Reduce any derived unit to these.
- Homogeneity: matching base units can rule an equation out, never in.
- Prefixes: work in powers of ten; the kilogram already carries a prefix.
One picture every idea returns to
A physical quantity has a magnitude and a unit. Seven SI base quantities, each with one base unit, generate every other unit by multiplication and division. Because both sides of a correct equation must reduce to the same base units, dimensional homogeneity is a fast first test of whether an equation can be right.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, A number is never enough.. Write "the speed is 30" and you have said almost nothing. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Weigh the units, not the numbers | Reduce each side of an equation to SI base units and put them on a balance. Project the simulation and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: The base, the derived, the prefix | Five base quantities carry almost all of AS mechanics and electricity. | Pairs |
| 21 to 30 min | Teach and check: Estimates the examiner expects | Estimates the examiner expects. | Connect-Extend-Challenge |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 15 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Estimates the examiner expects, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Connect-Extend-Challenge
After the new idea, each student writes three lines: how it connects to what they knew, how it extends their thinking, and what still challenges them. Pairs compare, then the challenges become the plenary.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Do not treat a successful homogeneity check as proof. | Units cannot detect a missing pure number such as the ½ in Eₖ = ½mv² or a factor of 2π, because pure numbers have no units. A homogeneous equation can still be wrong. The check only ever does one thing with certainty: if the base units do not match, the equation cannot be correct. Also watch the kilogram, the only base unit that already carries a prefix; the base unit of mass is the kilogram, not the gram. |
Physical quantities and units: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
State what is meant by a physical quantity, and use the quantity "a current of 0.50 A" to identify its two parts.
The joule is the SI unit of energy. Starting from a suitable defining equation, express the joule in terms of SI base units.
The drag force on a sphere falling through a liquid is modelled by F = k r v, where r is a radius, v is a speed and k is a constant. Determine the SI base units of k.
A wavelength is recorded as 0.0000006 m. Express this in nanometres and state the prefix used.
Estimate, to the nearest order of magnitude, the number of seconds in a human lifetime. Show your reasoning and give the answer with its unit.
A student checks the equation v² = u² + 2as by units and finds it homogeneous. They conclude the equation must be correct. Explain why their reasoning is flawed, and state what the check does establish.
Answer key and marking notes
- A physical quantity is a magnitude (a numerical value) together with a unit. ✓
- For 0.50 A the magnitude is 0.50 and the unit is the ampere (A). ✓
- Use energy = force × distance, so 1 J = 1 N × 1 m. ✓
- The newton in base units is kg m s⁻². ✓
- So 1 J = kg m s⁻² × m = kg m² s⁻². ✓
- Rearrange: k = F / (r v). ✓
- Units: F is kg m s⁻², r is m, v is m s⁻¹. So k = (kg m s⁻²) / (m × m s⁻¹). ✓
- This simplifies to kg m⁻¹ s⁻¹. ✓
- 0.0000006 m = 6 × 10⁻⁷ m = 600 × 10⁻⁹ m. ✓
- Since nano means 10⁻⁹, this is 600 nm. ✓
- One year is about 3 × 10⁷ s (365 × 24 × 3600 ≈ 3.15 × 10⁷). ✓
- A lifetime of order 80 years gives about 80 × 3 × 10⁷ ≈ 2.5 × 10⁹, so of order 10⁹ s. ✓
- Homogeneity is necessary but not sufficient: a units check cannot detect a wrong dimensionless factor (for example a missing or extra pure number), so a homogeneous equation can still be wrong. ✓
- What it does establish: if the units had not matched, the equation would certainly be wrong. A successful check only shows the equation is not ruled out by units. ✓
- (Here each term reduces to m² s⁻², so it passes the test, which is consistent with the equation being correct but does not prove it.) ✓
18 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Which quantity is measured in an SI base unit?
- AMass
- BEnergy
- CForce
- DDensity
Which of these is not an SI base unit?
- AThe metre
- BThe ampere
- CThe newton
- DThe second
The SI base unit of thermodynamic temperature is the:
- Ajoule
- Bkelvin
- Ckilogram
- Ddegree Celsius
The prefix micro (µ) represents a factor of:
- A10⁻²
- B10⁻⁹
- C10⁻³
- D10⁻⁶
A power of 1.0 MW expressed in watts is:
- A1.0 × 10⁶ W
- B1.0 × 10³ W
- C1.0 × 10⁻⁶ W
- D1.0 × 10⁹ W
A time interval of 5.0 ms is the same as:
- A5.0 × 10³ s
- B5.0 × 10⁻² s
- C5.0 × 10⁻³ s
- D5.0 × 10⁻⁶ s
The newton, expressed in SI base units, is:
- Akg m² s⁻²
- Bkg m s⁻²
- Ckg m s⁻¹
- Dkg m⁻¹ s⁻²
The joule, expressed in SI base units, is:
- Akg m s⁻²
- Bkg m² s⁻³
- Ckg m² s⁻¹
- Dkg m² s⁻²
The watt, expressed in SI base units, is:
- Akg m² s⁻³
- Bkg m² s⁻¹
- Ckg m² s⁻²
- Dkg m s⁻³
The coulomb, expressed in SI base units, is:
- AA s⁻¹
- BA² s
- CA s
- Dkg A s
The volt is energy per unit charge. In SI base units it is:
- Akg m s⁻³ A⁻¹
- Bkg m² s⁻³ A⁻¹
- Ckg m² s⁻³ A
- Dkg m² s⁻² A⁻¹
The SI base units of density are:
- Akg m³
- Bkg m⁻²
- Ckg m⁻¹ s⁻²
- Dkg m⁻³
Which statement about a homogeneous equation is correct?
- ABoth sides reduce to the same combination of base units
- BBoth sides have the same numerical value
- CIt is guaranteed to be physically correct
- DIt contains no constants
In v = k √(P / ρ), where v is a speed, P a pressure and ρ a density, the constant k is:
- Ameasured in m s⁻¹
- Bmeasured in kg
- Cdimensionless
- Dmeasured in pascals
Which quantity has the same SI base units as energy?
- Apressure × length
- Bforce × distance
- Cforce × time
- Dmass × velocity
The product (force × time) has the same SI base units as:
- Aenergy
- Bpower
- Cpressure
- Dmomentum
A reasonable estimate of the mass of a medium apple is:
- A0.1 kg
- B5 kg
- C0.5 g
- D1.0 kg
A reasonable estimate of the diameter of a human hair is:
- A1 mm
- B1 m
- C0.1 mm
- D1 cm
Quiz key
Mass has the base unit kilogram. Force (N), energy (J) and density (kg m⁻³) are all derived from base units.
The newton is a derived unit (kg m s⁻²). Metre, second and ampere are base units.
Thermodynamic temperature is measured in kelvin (K). The degree Celsius is a derived scale offset from it.
Micro means 10⁻⁶. Milli is 10⁻³ and nano is 10⁻⁹.
Mega means 10⁶, so 1.0 MW = 1.0 × 10⁶ W.
Milli means 10⁻³, so 5.0 ms = 5.0 × 10⁻³ s.
Force = mass × acceleration = kg × m s⁻² = kg m s⁻².
1 J = 1 N × 1 m = kg m s⁻² × m = kg m² s⁻².
1 W = 1 J s⁻¹ = kg m² s⁻² × s⁻¹ = kg m² s⁻³.
Q = It, so the coulomb is the ampere second, A s.
V = J C⁻¹ = (kg m² s⁻²) / (A s) = kg m² s⁻³ A⁻¹.
Density = kg / m³ = kg m⁻³.
Homogeneity means matching base units on each side. It does not check numerical values or hidden dimensionless factors.
P/ρ has units (kg m⁻¹ s⁻²)/(kg m⁻³) = m² s⁻², whose square root is m s⁻¹, matching v. So k carries no units.
Force × distance is work, with units kg m² s⁻², the same as energy. Force × time is impulse, and mass × velocity is momentum.
Force × time = kg m s⁻² × s = kg m s⁻¹, the units of momentum (mass × velocity).
A medium apple is about 100 g = 0.1 kg. One kilogram would be a very large apple.
A human hair is roughly 0.1 mm across (order 10⁻⁴ m). One millimetre would be very thick.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.