Help students see dimensions as the "type" of quantity (length, time, mass), distinct from the numbers and units themselves. This is roughly Lessons 1-2.
ConceptualIntroduce [L], [T], [M]
Step 1 - Introduce dimensions vs. units
Start with concrete quantities and ask what they have in common (e.g., 5 m, 3 s, 10 kg, 25 mi/h). Lead students to the idea that each is "of a certain type" with an associated unit.
Dimension = nature of quantity (e.g. length, time, mass).
Unit = how we measure that dimension (meters, seconds, kilograms).
Addition/subtraction only makes sense for same dimensions.
Activity: Put "5 m + 3 s" on the board and ask why it doesn't make sense.
Step 2 - Bracket notation
Introduce shorthand notation for dimensions:
[L] = length
[T] = time
[M] = mass
[Theta] = temperature (optional)
Quick tasks: Ask students for the dimensions of distance, time, speed, acceleration, and force; have them write in [ ]-notation.
Phase 2 Procedural Skills
Turn the conceptual ideas into a systematic procedure for checking equations, deriving relationships, and converting units (Lessons 3-4).
ProceduralGolden rule: homogeneity
Step 3 - Dimensional homogeneity
Emphasize the golden rule: every valid physical equation must be dimensionally homogeneous.
Left and right-hand sides must have the same dimensions.
Terms added or subtracted must share dimensions.
Example: v = at => [L][T]^-1 = [L][T]^-2 x [T].
Activity: Give several equations and have students label them "OK / not OK" based on dimensions alone.
Step 4 - Dimensional analysis procedure
Teach a repeatable algorithm students can apply:
A. Identify what you're solving for and its dimensions.
B. List all given quantities and their dimensions.
C. Combine them (via multiplication/division) to match target dimensions.
D. Check dimensional consistency.
E. Substitute values and compute.
Example: Using m and a, find a formula for force with dimensions [M][L][T]^-2.
Step 5 - Guided examples
Walk through canonical examples together:
Checking v = at for consistency.
Deriving pendulum period T ~ sqrt(L/g) by solving exponent equations.
Simple unit conversion, e.g. 60 mph -> m/s.
Try: Ask students to set up the exponent equations for T = kL^a g^b and solve for a and b.
Phase 3 Applications & Problem-Solving
Push dimensional analysis into real problem contexts: error checking, scaling arguments, and deriving relationships (Lessons 5-7).
ApplicationsError-check / Scale / Derive
Step 6 - Unit checking as error detection
Use units to catch mistakes the moment they're made. Every time students compute a new quantity, they should check its unit/dimensions.
Example: speed = distance / time => [L]/[T].
Spot nonsense like m^2/s x s instead of m/s.
Activity: Give "broken" worked examples with incorrect units and ask students to find where the mistake must have happened.
Step 7 - Scaling arguments
Show how dimensional thinking supports "what if we double the size?" reasoning.
Surface gravity scaling: g ~ M/r^2, with M ~ r^3 => g ~ r.
Students determine how g changes for a planet with 2x or 3x radius, same density.
Prompt: "If we triple the radius at constant density, how does g change?"
Step 8 - Deriving relationships from first principles
Use dimensional analysis to guess the shape of formulas before deriving them rigorously.
Example: kinetic energy E depending on m and v.
Set [E] = [M][L]^2[T]^-2 = [M]^a[L]^b[T]^-b and solve for a, b.
Conclude E ~ mv^2 (actual formula has dimensionless factor 1/2).
Challenge: Ask students to derive a possible form for a drag force depending on fluid density, speed, and area.
Phase 4 Mastery & Troubleshooting
Consolidate skills, highlight typical pitfalls, and practice across a range of problem types (Lesson 8+).
MasteryAvoid common traps
Step 9 - Common mistakes
Forgetting dimensionless constants (e.g. 1/2 in E = 1/2 mv^2).
Mixing units inside a calculation (meters with feet, hours with seconds).
Use this bank as a self-check for students after completing all phases, or as a pool for quizzes and exams. Each question has a collapsible suggested answer.
ReviewConcept + Skills
Students can jot an answer in the box, then click "Show answer" to compare with the suggested response. Use "Reveal all answers" if you'd like to project the key.
