What number expresses 3 m in centimetres?

300. Each metre contains 100 centimetres; the length itself stays unchanged.

Readings average 12.6 mm but are always 0.2 mm high. What is the corrected mean?

12.4 mm. Subtract the fixed excess; averaging has not removed it.

Using the even-digit convention, what is 2.745 rounded to three significant figures?

2.74. The exact halfway tie rounds toward the result ending in an even digit.

50 ± 1 cm has 2% uncertainty. What percentage uncertainty does 10 ± 1 cm have?

10%. The same absolute uncertainty is a larger fraction of the smaller measurement.

What dimensions remain when arc length is divided by radius?

None. Both lengths have dimension L, which cancels in their ratio.

Which proposed triangle areas pass the dimension check?

Both squared expressions. Both have area dimensions, although only half the square gives the triangle’s area.

What is 1 cm³ expressed in m³?

10⁻⁶ m³. Each of the three centimetre factors contributes a factor of 10⁻².

With length and gravity unchanged, what happens to the period when bob mass doubles?

It stays unchanged. Mass has exponent zero in the assumed period formula.

Teach Units and Measurement in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather a spoon, ruler, five matching coins, square paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

A length has a larger number in centimetres than in metres.
Final zeros in decimal measurements show how precisely a quantity was measured.
Nearby scenery appears to move faster than distant scenery during the same sideways movement.

What to teach

How to teach it

Making measurements

Steel spoon beside a ruler on a wooden tabletop.

1. Measurement and SI units

A measurement gives a number alongside an agreed unit.

Show: Place a spoon beside a ruler, with one end at zero. Read its length in centimetres. Divide that number by 100 to express the same length in metres.
Ask: Will using centimetres make the spoon’s measured number larger?
Say: Yes, smaller units need larger numbers for the same quantity.
Used in: Laboratories use specialised equipment to establish mass standards from the kilogram’s stable definition.

If your child says…“Changing the unit leaves the number unchanged.”

Ask: Would smaller units need more repetitions to span this spoon?
What’s true: The number changes to compensate for the unit’s size.

Five matching coins stacked beside an upright ruler.

2. Indirect measurement and instrument precision

Use measurable sizes to infer sizes too small to read directly.

Show: Stack five matching coins beside an upright ruler. Align zero with the stack’s base. After the prediction, measure the stack’s height.
Ask: Which will be easier to measure: one coin’s thickness or the whole stack’s height?
Say: The whole stack’s height is easier to measure. Divide the stack’s height by five to estimate the average coin thickness.
Used in: Magnified hair images reveal thickness when image width is divided by magnification.

If your child says…“Averaging removes a zero error.”

Ask: If every reading is too high, how would averaging remove that extra amount?
What’s true: A fixed offset remains in the average.

Reporting measurements

Number-line diagram showing an exact halfway value and a value just beyond halfway.

3. Significant figures and rounding

Report measured precision and round only after keeping extra calculation digits.

Show: Find 2.745 and 2.7451 on the number-line diagram. Compare their distances from the two endpoints.
Ask: Which endpoint will be nearer to 2.7451?
Say: 2.7451 is nearer 2.75, so it rounds to 2.75. Exactly halfway, choose the even final digit: 2.745 rounds to 2.74.
Used in: Decimal measurements retain final zeros to communicate their precision.

If your child says…“A final five always rounds upward.”

Ask: Which possible result ends in an even digit?
What’s true: At an exact halfway tie, this convention makes the last retained digit even.

$\frac{0.1}{20.0}\times100\%=0.5\%$

$\frac{0.1}{10.0}\times100\%=1.0\%$

$(20.0\times10.0)\,\mathrm{cm}^2=2.00\times10^2\,\mathrm{cm}^2$

4. Significant-figure arithmetic and relative uncertainty

Measurements limit how precisely we can report calculated results.

Show: Write these lines. Each rectangle edge has an assumed uncertainty of ±0.1 cm.
Ask: What approximate maximum percentage uncertainty will the area have?
Say: It is 1.5%, because percentage uncertainties add. The area keeps three significant figures, matching both lengths.
Used in: Calculated density retains only the precision supported by its mass and volume measurements.

If your child says…“The same uncertainty means the same percentage.”

Ask: Would the same uncertain length occupy more of the shorter edge?
What’s true: Percentage uncertainty equals uncertainty divided by measured value, multiplied by 100.

Using dimensions

Circle showing equal arc and radius lengths and their ratio defining the central angle.

5. Dimensional formulae and dimensionless angles

Dimensions track base quantities; ratios can cancel every dimension.

Show: Look at the diagram’s angle θ, arc length s and radius r. Compare the equal arc and radius, then read the angle’s ratio.
Ask: Will this angle have a length dimension, even though we measure it in radians?
Say: The angle is one radian because its arc length equals its radius. The length dimensions cancel, but radian remains the named unit.
Used in: Surveyors measure angles to find distances without crossing the ground between points.

If your child says…“An angle has the dimension of length.”

Ask: What length dimension survives dividing one length by another?
What’s true: Arc length and radius both have dimension L, so their ratio has no length dimension.

Square paper opened flat with a diagonal crease separating equal triangular halves.

6. Dimensional homogeneity

An equation is wrong if its two sides have different dimensions.

Show: Fold square paper diagonally so its halves overlap, then open it. Call the side length s. Compare the side length, its square and half its square as possible triangle areas.
Ask: Will checking dimensions distinguish the two expressions containing s²?
Say: Each triangle takes half the square, but both squared expressions have area dimensions. Matching dimensions cannot prove a formula.
Used in: Energy formula checks reject expressions whose dimensions do not match energy.

If your child says…“Matching dimensions proves the formula.”

Ask: What dimension could the factor ½ change?
What’s true: Both squared expressions pass because their numerical factors have no dimensions.

$1\,\mathrm{g}=10^{-3}\,\mathrm{kg}$

$1\,\mathrm{cm}^3=(10^{-2}\,\mathrm{m})^3=10^{-6}\,\mathrm{m}^3$

$1.00\,\mathrm{g/cm^3}=\frac{1.00\times10^{-3}\,\mathrm{kg}}{10^{-6}\,\mathrm{m}^3}$

7. Converting powered and compound units

Apply each conversion factor’s power, keeping denominator factors below.

Show: Write these lines. Say the mass conversion goes above and the cubed length conversion below.
Ask: What will $1.00\,\mathrm{g/cm^3}$ become in kilograms per cubic metre?
Say: The volume conversion is cubed because volume contains three length factors. Dividing the converted mass by the converted volume gives one thousand kilograms per cubic metre.
Used in: Density tables use matching units so material densities can be compared.

If your child says…“Volume uses the same conversion factor as length.”

Ask: Does a cubic centimetre contain one length factor or three?
What’s true: A volume conversion cubes the length conversion factor.

$P=Cm^a l^b g^c$

$T=M^aL^{b+c}T^{-2c}$

$(a,b,c)=(0,\tfrac12,-\tfrac12)$

8. Deriving power laws and estimating scales

Match dimensional powers to constrain formulas and estimate size changes.

Show: Write these lines for small pendulum swings. Name period P, mass m, length l, gravitational acceleration g and unknown numerical factor C.
Ask: What period formula follows from these powers?
Say: Period follows the square root of length divided by gravitational acceleration: $P=C\sqrt{l/g}$ . Dimensions leave C unknown.
Used in: Pendulum clocks use a fixed pendulum length to keep a steady rhythm.

If your child says…“A heavier bob doubles the period.”

Ask: Does the period’s time dimension need any mass factor?
What’s true: If only length, gravity and mass matter, dimensional balance gives mass a zero power.

A real problem

The kilogram now uses a constant of nature.

Physical objects used as kilogram standards could change mass over time.

In 2019, the kilogram was redefined using the Planck constant.

The replacement standard is inherently stable and can be checked anywhere.

NIST CSIR–NPL UK Government Chemist

Balance weighing loose rice on an Indian grocery counter.

Concept Lab · Common misconception

The tempting mistake in Units and Measurement

Open standalone misconception page →

Do more decimal places guarantee accuracy?

Wrong

A reading of 10.50 cm must be accurate because it includes hundredths.

Why it feels right. I associate fine resolution with closeness to the true value.

Right

A precise measurement can still have a systematic error.

Worked example

The three readings form a tight cluster around the wrong value.

An instrument reads 10.49, 10.50 and 10.51 cm on a 10.00 cm standard.
The readings cluster tightly, but the average is 0.50 cm too high.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

What number expresses 3 m in centimetres?

300. Each metre contains 100 centimetres; the length itself stays unchanged.
Readings average 12.6 mm but are always 0.2 mm high. What is the corrected mean?

12.4 mm. Subtract the fixed excess; averaging has not removed it.
Using the even-digit convention, what is 2.745 rounded to three significant figures?

2.74. The exact halfway tie rounds toward the result ending in an even digit.
50 ± 1 cm has 2% uncertainty. What percentage uncertainty does 10 ± 1 cm have?

10%. The same absolute uncertainty is a larger fraction of the smaller measurement.
What dimensions remain when arc length is divided by radius?

None. Both lengths have dimension L, which cancels in their ratio.
Which proposed triangle areas pass the dimension check?

Both squared expressions. Both have area dimensions, although only half the square gives the triangle’s area.
What is 1 cm³ expressed in m³?

10⁻⁶ m³. Each of the three centimetre factors contributes a factor of 10⁻².
With length and gravity unchanged, what happens to the period when bob mass doubles?

It stays unchanged. Mass has exponent zero in the assumed period formula.

15 more questions with answers, 2 with pictures

Teach Units and Measurement in eight steps.

Why it matters