Rounding: decimal places, significant figures and error intervals
Rounding numbers to an appropriate degree of accuracy, and writing the error interval for a rounded measurement
About three lessons of 45 to 60 minutes
Every measurement is a little bit uncertain, and that uncertainty can be written down exactly
A ruler measures a pencil as '12 cm long', but that '12' almost certainly hides a small amount of rounding, the pencil could genuinely be 11.6 cm, or 12.4 cm, and still round to 12 cm on a ruler marked in whole centimetres. Rounding is not just a way to make numbers tidier, it is something that happens to EVERY real-world measurement, and being precise about HOW MUCH uncertainty a rounded number hides is a genuinely useful skill.
2 closely related rounding skills matter here: rounding a calculated number sensibly (to a number of decimal places, or to a number of significant figures, the first few 'meaningful' digits), and working the other way, taking an already-rounded measurement and writing down the full range of values (the ERROR INTERVAL) it could actually represent.
- A pencil measured as 12 cm (nearest cm)could really be anywhere from 11.5 cm up to (but not including) 12.5 cm
- A bag of flour labelled 1.5 kg (nearest 0.1 kg)could really be from 1.45 kg up to (but not including) 1.55 kg
- 3.14159... rounded to 2 decimal places3.14, keeping exactly 2 digits after the decimal point
- A population of 8,124,532 rounded to 2 significant figures8,100,000, keeping only the first 2 meaningful digits
What students will be able to do
Students will round numbers to a given number of decimal places or significant figures, and write the error interval for a measurement rounded to a given degree of accuracy, using inequality notation.
- I can round a number to a given number of decimal places, using the next digit to decide whether to round up.
- I can round a number to a given number of significant figures, correctly identifying the first significant digit even when the number starts with a decimal point.
- I can find the error interval for a rounded measurement, using inequality notation a <= x < b.
Standards this unit teaches
- KS3 Maths: NumberUK National Curriculum (England)Number
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Number" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures]"; "use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x<=b".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Rounding learn guidethe foundational rounding rule (look at the next digit) this unit extends to significant figures
- Grade 6 area and volume teaching unitreal-world measurement contexts where a sensibly-rounded answer matters
- Decimals learn guideplace value in decimal numbers, needed to identify decimal-place and significant-figure positions
Words to teach and display
- Decimal place
- a digit's position after the decimal point (1st, 2nd, 3rd, and so on)
- Significant figure
- one of a number's 'meaningful' digits, starting from the first non-zero digit, used to indicate how precisely a number is known
- Round up / round down
- increasing (or keeping) the last kept digit, decided by looking at the very next digit (5 or more rounds up)
- Error interval
- the full range of values a rounded measurement could actually represent, written using inequality notation
- Degree of accuracy
- how precisely a number has been rounded (e.g. to the nearest whole number, or to 2 decimal places)
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Rounding to a number of decimal places
ConcreteTo round to a given number of decimal places, keep exactly that many digits after the decimal point, and look at the NEXT digit to decide whether to round the last kept digit up. If that next digit is 5 or more, round up; if it is 4 or less, leave the last kept digit unchanged.
Round 7.4863 to (a) 1 decimal place, and (b) 2 decimal places.
- (a) Keep 1 digit after the point (the 4). Look at the next digit, 8, which is 5 or more, so round up: 7.4 becomes 7.5.
- (b) Keep 2 digits after the point (4, 8). Look at the next digit, 6, which is 5 or more, so round up: 7.48 becomes 7.49.
Answer: (a) 7.5. (b) 7.49.
- Why does 7.4863 round DIFFERENTLY to 1 dp than to 2 dp, even though it's the same original number?
- Round 12.376 to 2 decimal places.
2. Rounding to significant figures
PictorialThe FIRST significant figure is always the first non-zero digit, reading from the left, no matter where the decimal point is. 0.00456 has its 1st significant figure at the 4 (the leading zeros do not count). After the first significant figure, every digit counts as significant, including zeros between other digits.
Round 3842 to (a) 1 significant figure, and (b) 3 significant figures. Round 0.00728 to 1 significant figure.
- (a) The 1st significant figure of 3842 is 3. Look at the next digit, 8, round up: 3842 rounds to 4000 (to 1 s.f.).
- (b) The first 3 significant figures are 3, 8, 4. Look at the next digit, 2, round down (keep as is): 3842 rounds to 3840 (to 3 s.f.).
- 0.00728: the leading zeros are not significant, so the 1st significant figure is 7. Look at the next digit, 2, round down: 0.00728 rounds to 0.007 (to 1 s.f.).
Answer: 3842 to 1 s.f. = 4000. 3842 to 3 s.f. = 3840. 0.00728 to 1 s.f. = 0.007.
- Why don't the leading zeros in 0.00728 count as significant figures?
- Round 592.7 to 2 significant figures.
3. Error intervals from a rounded measurement
AbstractIf a measurement has been rounded to the nearest whole unit, the TRUE value could be anywhere up to HALF a unit above or below the rounded value. That range is the error interval, written as a lower bound (inclusive, using <=) up to an upper bound (exclusive, using <), because a value exactly at the upper bound would actually round UP to the next rounded value instead.
A mass is measured as 40 kg, rounded to the nearest 10 kg. Write the error interval for the true mass, x.
- The rounding unit is 10, so half of it is 5.
- Lower bound: 40 - 5 = 35. Upper bound: 40 + 5 = 45.
Answer: 35 <= x < 45.
- Why is the upper bound written with a strict < instead of <=?
- A length is measured as 6.2 cm, rounded to the nearest 0.1 cm. Write its error interval.
Common misconceptions and how to address them
MisconceptionSignificant figures and decimal places are the same thing.
Why it happens: Both involve 'keeping some digits and rounding the rest', so the 2 ideas blur together without a clear worked contrast.
How to address it: Show the SAME number rounded both ways side by side, e.g. 0.0384 to 2 decimal places (0.04) versus 0.0384 to 2 significant figures (0.038), making the difference between COUNTING FROM THE DECIMAL POINT and COUNTING FROM THE FIRST MEANINGFUL DIGIT explicit.
MisconceptionThe leading zeros in a number like 0.0056 count as significant figures.
Why it happens: Every digit LOOKS equally important on the page, so there's no visual cue that leading zeros are just placeholders rather than measured precision.
How to address it: Explain that leading zeros only fix the position of the decimal point, they carry no information about how precisely the number was actually measured; only digits from the first non-zero digit onward do.
MisconceptionThe error interval for a value rounded to 20 (nearest 10) is written as 15 <= x <= 25 (using <= on both sides).
Why it happens: Using <= symmetrically feels natural, but a true value of exactly 25 would round to 30 (not 20), so 25 cannot correctly be included in the interval for 20.
How to address it: Test the boundary value explicitly: check which rounded value 25 itself would actually produce (30, not 20), proving the upper bound must be a strict < to correctly exclude it.
Guided practice (with answers)
1. Round 5.6274 to 2 decimal places.
Answer: 5.63, because the next digit (7) rounds the 2nd decimal place up.
2. Round 8.041 to 1 decimal place.
Answer: 8.0, because the next digit (4) rounds down.
3. Round 6753 to 1 significant figure.
Answer: 7000, because the next digit (7) rounds the leading 6 up to 7.
4. Round 0.0294 to 2 significant figures.
Answer: 0.029, because the 1st 2 significant figures are 2 and 9, and the next digit (4) rounds down.
5. A length is measured as 8 cm, rounded to the nearest cm. Write the error interval.
Answer: 7.5 <= x < 8.5.
6. A mass is measured as 250 g, rounded to the nearest 10 g. Write the error interval.
Answer: 245 <= x < 255.
Independent practice worksheets
Practise rounding to decimal places, significant figures, and finding error intervals with computed, never-wrong answer keys.
Differentiation
- Underline the digit being kept and circle the 'decider' digit immediately after it, every single time, before deciding to round up or down.
- For significant figures, cross out leading zeros first so only genuinely significant digits remain visible to count.
- For error intervals, always compute 'half the rounding unit' as its own explicit step before finding the bounds.
- Start error intervals with whole-number rounding units (nearest 1, nearest 10) before introducing decimal rounding units (nearest 0.1).
- Round the SAME number to several different degrees of accuracy (1 dp, 2 dp, 1 sf, 2 sf, 3 sf) and compare how the results differ.
- Investigate a case where rounding to a number of decimal places and to the same COUNT of significant figures gives the same result, and explain why.
- Given an error interval, work out what degree of accuracy the original rounding used.
- Explore how error intervals combine when 2 rounded measurements are added together (the combined error can be bigger than either 1 alone).
Assessment: exit ticket
A three-question exit ticket sampling decimal places, significant figures and error intervals.
1. Round 14.638 to 1 decimal place.
Answer: 14.6, because the next digit (3) rounds down.
2. Round 47,382 to 2 significant figures.
Answer: 47,000, because the next digit (3) rounds down.
3. A length is measured as 3.5 cm, rounded to the nearest 0.1 cm. Write the error interval.
Answer: 3.45 <= x < 3.55.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 decimal places (section 1), Lesson 2 significant figures (section 2), Lesson 3 error intervals (section 3) plus the exit ticket.
- Every rounding answer in the matching worksheets is computed via pure integer arithmetic on the number's digit string (never a raw floating-point .toFixed on an uncontrolled value), so rounding is exact even in edge cases; any generated case where rounding would carry into an extra digit (e.g. 995 to 2 s.f. carrying to 1000, a different digit count) is detected and re-generated rather than risking a malformed answer.
- priorKnowledge deliberately avoids linking to the UK KS3 batch-6 fractions/decimals unit (which would otherwise be the ideal decimal-fluency prerequisite): it is not yet merged into main, and linking to it here would create a broken link on this branch, the same failure mode batch 6 itself hit and fixed for its own priorKnowledge links.