A ruler reading of 10.2 cm is not the same as knowing an exact length of 10.200000 cm. If the ruler reports to the nearest 0.1 cm and rounding is the only source of error, the original length is within 0.05 cm of the reading. Start by writing that interval before multiplying measurements or choosing how many digits to report.
Predict
Predict whether a calculator displaying many digits creates more measurement information. It does not. Predict also whether half an instrument’s smallest displayed increment always bounds the true error. That only describes rounding under an assumed calibration; a shifted zero or inconsistent reading adds another source of uncertainty. This lesson uses explicit interval assumptions rather than invented confidence percentages.
Experiment
A rectangle has reported length 10.2 cm and width 4.6 cm, each rounded to the nearest 0.1 cm. Use these conservative compatible bounds.
Quantity
Lower bound
Displayed value
Upper bound
Length, cm
10.15
10.2
10.25
Width, cm
4.55
4.6
4.65
Area, cm²
46.1825
46.92
47.6625
Multiply the lower sides and upper sides yourself. Both dimensions are positive, so area increases with either side. The midpoint-product 46.92 is one model estimate; it is not a guarantee that the actual area has two-decimal-place accuracy.
Understand
Resolution is the spacing between display levels or scale markings. Under rounding to the nearest increment r, a reading m is compatible with values between m−r/2 and m+r/2. A specified tie rule may include only one endpoint, while a conservative closed interval includes both. State which convention you use. A guaranteed error interval is also different from a statistical confidence interval: this example assigns no probability distribution to possible lengths.
For positive lengths L in [L_low,L_high] and W in [W_low,W_high], the area A=LW lies between L_low W_low and L_high W_high. This follows from multiplication preserving inequalities for nonnegative numbers. The rule cannot be copied unchanged to arbitrary intervals containing negative numbers; there all endpoint products must be compared. In a measurement context, check positivity before using the shortcut.
Reporting is a decision about what a reader may infer. A useful summary for the worked rectangle is about 47 cm², accompanied by the compatible interval approximately 46.18 to 47.67 cm² if outward-rounded bounds are wanted. Rounding the lower endpoint downward and the upper endpoint upward preserves containment. Writing the exact arithmetic bounds is also legitimate if they are clearly labeled as consequences of the assumed input bounds, not as direct instrument precision.
The exact area formula and the measured inputs play different roles. A perfectly correct formula does not remove input uncertainty. Likewise, significant-figure conventions can be a convenient reporting habit, but an explicit interval explains the actual claim and exposes cases where uncertainty is asymmetric or dominated by one input.
Apply
Compare two measured objects by their intervals rather than only their displayed centers. Overlap means the available bounds may permit either ordering; nonoverlapping bounds can certify an ordering under the assumptions. Choose a report for a class project that includes the measured quantity, units, resolution or bounds, calculated estimate and one sentence explaining the retained precision.
Continue
These activities address measurement-limited reporting in the school quantities standards; they do not certify a complete standards sequence. Next, choose the quantity itself and the axes used to display it. A precise number can still answer the wrong question if its denominator or units are inappropriate.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A ruler reports 7.4 cm rounded to the nearest 0.1 cm. Under rounding alone, what is the lower endpoint of the compatible interval?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A rectangle has side intervals [2.9,3.1] cm and [4.8,5.2] cm. What is the largest compatible area in square centimeters?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A display rounded to the nearest 0.2 kg shows 18.6 kg. State a compatible interval and defend a report that does not imply milligram accuracy.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A board length is 12.0±0.1 cm and width 5.0±0.1 cm, interpreted as inclusive guaranteed bounds. Give area bounds and choose an honest summary.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Two objects have mass intervals [9.8,10.2] g and [10.0,10.4] g. Can you certify which is heavier? Explain with compatible counterexamples.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A ruler has 1 mm markings but may be miscalibrated by 3 mm. Explain why ±0.5 mm is not a justified total error bound.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
