Design a measurement investigation using stacked cards

Design a measurement investigation using stacked cards

Draft audience: Grade 10 intended, subject to teacher review of prerequisites and exact current subject, strand and term placement. This is original learning support, not an official curriculum allocation or marking scheme.

A ruler can measure a stack of thin cards more usefully than one card that is thinner than its smallest division. The calculation still depends on counted cards, aligned edges, compression and the way the scale is read. This investigation connects an apparatus choice to repeated endpoint records and an honest uncertainty statement. All numerical readings are simulated; no measurement has actually been carried out.

Plan before calculating

Question: Is measured stack thickness approximately proportional to the number of nominally identical cards, and what average thickness per card is consistent with the supplied readings? The chosen counts are 20, 40 and 60 cards. The manipulated variable is card count; the measured variable is the difference between the ruler positions of the lower and upper stack surfaces, in millimetres.

The paper apparatus list is a ruler with 1 mm divisions, pre-counted stacks of one card type, a flat horizontal surface and a flat lightweight top sheet used consistently to indicate the upper surface. Its thickness is excluded from the stated stack readings. A physical version, if chosen separately by a teacher, needs only teacher-approved cards and a blunt ruler on a stable desk. No sharp cutting, heavy loads, elevated stacks or powered equipment is required.

Keep the card type, dryness, alignment and top-sheet pressure comparable. Read perpendicular to the scale to reduce parallax. Rebuild or reposition each stack for three independent readings rather than copying the same displayed value three times. Count each stack carefully: a wrong count can bias the calculated per-card result even when the ruler is read consistently.

Read the two endpoints

Complete diagram alternative: the lower surface is aligned with 12 mm and the upper card surface with 24 mm on the same ruler. Thickness is 24 − 12 = 12 mm. The lower reading is not required to be zero. Subtract matching endpoints; using 24 mm alone would count the distance below the stack as well.

Calculate a unit-aware result

For 40 cards, the mean is (12 + 13 + 12) ÷ 3 = 37/3 mm. Dividing by the exact counted number gives (37/3) ÷ 40 = 37/120 mm per card, about 0.308 mm. The additional digits show the arithmetic, not measurement certainty. A practical summary is approximately 0.31 mm per card under these stack conditions.

Adding 20 cards increases each successive mean by 6.00 mm, which corresponds to 0.300 mm per added card. The fitted relationship through these three means would have a small positive intercept of 1/3 mm, rather than passing exactly through zero. These rounded, varying readings are consistent with an approximately proportional pattern; they do not prove exact proportionality or a universal card thickness. Do not force the means to 6, 12 and 18 to make a cleaner line.

Separate resolution, repeatability and bias

Use the explicit rounding model that each endpoint rounded to the nearest millimetre differs from its underlying scale position by at most 0.5 mm. In a difference, one endpoint can round upward while the other rounds downward. The worst-case rounding contribution to a single thickness is therefore at most 1 mm. This is a bound for the stated model, not the total measurement uncertainty.

Dividing that bound by 20 gives 0.050 mm per card; by 40 gives 0.025 mm; by 60 gives about 0.0167 mm. The same absolute endpoint limitation has a smaller per-card effect for a larger counted stack. Do not automatically divide this worst-case bound by the square root of three simply because three repeats exist; no probabilistic error model has been supplied.

Each stack’s observed range is 1 mm. That describes repeatability, not proof of accuracy. A stretched ruler scale would bias differences in a way repetition cannot remove. By contrast, a fixed additive scale offset applied equally to both endpoints cancels when they are subtracted. Gaps, curled cards or changing pressure can also affect the stack. Larger stacks help with resolution but cannot guarantee those effects disappear.

Independent measurement check

A new 50-card stack has lower/upper endpoint pairs (11, 26), (13, 29) and (9, 24), all in mm. The same nearest-millimetre rounding model applies.

1. Calculate the three thicknesses, their mean and range.
2. Estimate average thickness per card and convert it to centimetres.
3. Find the single-reading worst-case rounding contribution per card.
4. A fourth record has unknown lower position and upper position 27 mm. Can it be added to the mean as 27 mm? Explain.

Answers and review criteria

The differences are 15, 16 and 15 mm (3 points). Mean = 46/3 ≈ 15.33 mm and range = 1 mm (2 points). Mean per card = 46/150 ≈ 0.307 mm = approximately 0.0307 cm (2 points). The rounding contribution is at most 1/50 = 0.020 mm per card under the stated model (1 point). The missing lower endpoint prevents a thickness calculation; 27 mm is a position, not an established thickness (2 points). Total: 10 learning points.

Finish by stating which limitation a proposed improvement addresses. A finer scale addresses resolution; consistent pressure addresses compression differences; an independent scale check addresses calibration. More decimal places address none of those by themselves. OpenStax’s measurement and uncertainty discussion supports the foundational distinction; the stacks, diagram, rounding model and calculations here are original.