aiMOOC

Rounding significant figures and bounds - Mathematics Year 9

QR-Code

Introduction

Does “two figures” mean two decimal places or two significant figures? Track place value, choose a precision and use the reported accuracy to describe possible original values.

Complete eight rounding tasks and four bound intervals. Half-open bounds here use a stated positive-number round-half-up convention; this editorial GCSE bridge is not labelled a separate statutory Year 9 requirement.

Complete eight place-value rounding decisions

Decimal places count positions after the decimal point; significant figures start at the first nonzero digit and count meaningful digits from there. Thus 3.46 to two significant figures is 3.5, because the hundredths digit 6 rounds the tenths digit 4 upward. The same 3.46 to two decimal places remains 3.46, already reported at that accuracy.

For positive values, use round-half-up throughout this course: nearest values are selected, and an exact halfway tie goes upward. Other tie conventions exist, so state the convention before making endpoint claims. Our examples and bounds do not silently generalise this positive-number convention to negative tie cases.

Eight rounding tasks. The rounding unit states the original place being rounded. In row 7, the carrying creates a two-digit number, but 10.0 preserves the intended tenths accuracy.

Task	Original value	Required accuracy	Rounding unit	Reported result
1	3.46	2 significant figures	0.1	3.5
2	3.46	2 decimal places	0.01	3.46
3	0.004876	2 significant figures	0.0001	0.0049
4	0.004876	3 decimal places	0.001	0.005
5	7865	2 significant figures	100	7900
6	7865	nearest thousand	1000	8000
7	9.96	nearest 0.1	0.1	10.0
8	12.345	2 decimal places	0.01	12.35

For 0.004876, the leading zeros locate the decimal size; they do not start significant-figure counting. The first two significant digits are 4 and 8, the next 7 rounds 8 to 9, so the result.0049. To three decimal places, the first three fractional positions are 0, 0, 4 and the next 8 rounds the thousandths digit upward, giving.005. The two requests therefore produce different results.

For 7865, two significant figures retains the thousands and hundreds positions. The next digit 6 rounds the hundreds 8 to 9, giving 7900. Write 7.9×10³ if the significant-figure precision could be ambiguous in a whole-number display. Nearest thousand instead retains the thousands position and rounds to 8000; that is a different rounding unit.

For 9.96 to the nearest 0.1, the retained place is tenths. The next digit 6 produces a carry: 9.9 rounds to 10.0. This is explicitly a fixed-place request. The result 10.0 has three significant figures; it must not be described as a two-significant-figure display. Normalised 1.0×10¹ has two significant digits and states ones-level precision instead. Write the actual requested accuracy rather than silently changing it after a carry.

For 12.345 to two decimal places, the first omitted digit 5 is an exact tie at the hundredths decision under our convention, and the answer 12.35 follows. Do not round first to three significant figures and then to decimal places: successive rounding can change a final result. Keep the original value until the requested final precision.

Explain four half-open rounding intervals

A rounded report usually represents many possible original values. If positive x is rounded to a fixed unit u and reported as r under round-half-up, then $r - \frac{u}{2} \leq x < r + \frac{u}{2}$ . This interval describes rounding alone, not every possible instrument, calibration or sampling error.

Four complete bound intervals. Use the row's unit for both endpoints and retain the lower-inclusive, upper-exclusive signs.

Task	Reported value	Stated rounding unit	Lower included	Upper excluded
1	18 cm	Nearest 1 cm	17.5 cm	18.5 cm
2	3.5 kg	Nearest 0.1 kg	3.45 kg	3.55 kg
3	120 s	Nearest 10 s	115 s	125 s
4	0.0060 m	Nearest 0.0001 m	0.00595 m	0.00605 m

For the source 18 cm rounded to nearest centimetre, the interval is 17.5 cm≤x<18.5 cm. Lower endpoint 17.5 is halfway between 17 and 18 and rounds upward to 18. Upper endpoint 18.5 is halfway between 18 and 19 and rounds upward to 19, so it is excluded. It is a least upper bound, not an attainable maximum for values rounding to 18 under this convention.

For 3.5 kg to nearest.1 kg, half a unit is.05 kg, giving 3.45 kg≤x<3.55 kg. For 120 s to nearest 10 s, half a unit is 5 s, giving 115 s≤x<125 s. Do not take half of the reported 120; the half-width comes from the rounding unit 10, not the central value.

For.0060 m to nearest.0001 m, half-width.00005 m gives.00595 m≤x<.00605 m. The last zero records stated accuracy here. Write.0001 explicitly rather than inferring precision from an unlabeled string of trailing zeros. Reporting.006 to fewer places would represent a wider interval under a different rounding unit.

A labelled number-line interpretation for the original length is:

Source length reported18cm,nearest1cm
17.5cm  [----------------------)  18.5cm
         included                 excluded
Every interior value rounds to18cm.
18.5cm itself rounds to19cm.

Check a value just below the upper endpoint, such as 18.499 cm: it rounds to 18 cm. It can be made closer still to 18.5 without ever reaching that excluded endpoint. This explains why writing “maximum possible 18.5 cm” can mislead; state the upper bound and strict inequality instead.

Estimate, check endpoints and transfer units

Estimate calculations deliberately. Rounding 19.8×4.12 to one significant figure per factor gives 20×4=80. The exact product is 81.576, so the estimate is plausible and close. For 48.6÷2.05, use 50÷2=25; the exact quotient is about 23.7073. An estimate checks scale rather than promising the requested final decimal accuracy.

Choose convenient rounding based on the purpose. For a tight budget limit, an estimate near the limit cannot decide feasibility; return to exact values or a justified bound. For checking whether a calculator entry is off by a factor of ten, one significant figure may be enough. Report which inputs were rounded and avoid presenting the estimate as an exact equality to the original calculation.

The original Corbettmaths decimal-place tutorial rounds 8.32 to 8.3 and.89 to.9 at one decimal place. Compare that fixed-place method with our significant-figure tasks, then explain why it does not by itself determine a bound interval without a stated rounding unit and tie convention. Original: corbettmaths, Rounding to 1 and 2 Decimal Places - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Independent inverse check for an interval. The midpoint of 17.5 and 18.5 is 18, and their difference is 1 cm, the original rounding unit. For the.0060 m interval, the endpoints differ by.00010 m and average to.00600 m. Reconstructing centre and width is a second check on adding or subtracting half a unit.

Unfamiliar transfer. A positive mass reported 2.40 kg to nearest.01 kg lies in 2.395 kg≤m<2.405 kg. The lower endpoint rounds upward to 2.40 and the upper to 2.41. Converting all values to grams gives 2395 g≤m<2405 g, with a 10 g rounding unit. A correct unit conversion scales both the centre and half-width.

First-error analysis. Writing 17≤x<19 for the source length uses a whole rounding unit on either side instead of half a unit. Writing 17.5≤x≤18.5 includes the tie that rounds to 19. Writing.004876≈.00 to two significant figures starts counting from leading zeros rather than the first nonzero digit. Identify the first incorrect choice before repairing the final number.

Eight correctly copied answers are not the whole goal. Retain the requested accuracy, original rounding unit, one original-value or place-value check, four signed intervals and the endpoint explanation. This course treats measurement reports as supplied mathematical data; it does not certify an actual instrument reading or claim to account for all physical uncertainty.

Interactive Tasks

Quiz: Test Your Knowledge

3.46 to 2 significant figures? (3.5) (!3.46) (!3.4) (!3.6)

0.004876 to 2 significant figures? (0.0049) (!0.00) (!0.0048) (!0.0050)

7865 to 2 significant figures? (7900) (!8000) (!7860) (!7870)

9.96 rounded to nearest 0.1? (10.0) (!9.9) (!9.96) (!10.1)

Positive 12.345 to 2 decimal places, half-up? (12.35) (!12.34) (!12.3) (!12.30)

18 cm to nearest 1 cm: lower included? (17.5 cm) (!17 cm) (!18 cm) (!18.5 cm)

Same report: 18.5 cm is? (Excluded because it rounds to 19 cm) (!Included because it is the upper bound) (!The only original value) (!The rounding unit)

120 s to nearest 10 s: upper excluded? (125 s) (!130 s) (!120 s) (!115 s)

0.0060 m to nearest 0.0001 m: lower included? (0.00595 m) (!0.0059 m) (!0.00605 m) (!0.005 m)

19.8×4.12 estimated with 1 sf per factor? (80) (!81.576) (!800) (!8)

Memory Game

Decimal places	Counted fractional positions after the point
Significant figures	Count beginning at the first nonzero digit
Rounding unit	Place increment at which a value is rounded
Lower bound	Included endpoint under stated positive half-up convention
Upper bound	Excluded endpoint under stated positive half-up convention
Estimate	Approximation used for a stated purpose

Drag and Drop

3.5	Source 3.46 to 2 significant figures
0.0049	Small value to 2 significant figures
7900	7865 to 2 significant figures
17.5 cm	Source lower included bound
18.5 cm	Source upper excluded bound

Crossword Puzzle

ROUNDING	Reporting at a selected accuracy
SIGNIFICANT	Figures counted from first nonzero
DECIMAL	Places counted after the point
INTERVAL	Set between stated endpoints
ESTIMATE	Approximate calculation
PRECISION	Detail of a stated numerical report

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Distinguish two accuracy requests: Compare 3.46 to 2 sf and 2 dp.
Locate the first nonzero digit: Read.004876 before deciding which digit controls rounding.
Read the source interval: Explain both endpoints and their tie outcomes.
Name the rounding unit: Separate nearest 10 s from the central value 120 s.

Standard

Complete eight rounding tasks: Retain original values, requested accuracies and units.
Write four complete intervals: Use correct endpoints, units and inequality signs.
Estimate two calculations: Compare 80 with 81.576 and 25 with about 23.7073.
Check centre and width: Recover each reported value and rounding unit from endpoints.

Advanced

Explain carrying at a threshold: Distinguish 10.0 at nearest.1 from ambiguous significant-figure display.
Transfer to a new unit: Convert the 2.40 kg interval and its width into grams.
Repair first errors: Correct leading-zero counting, whole-unit half-width and included-upper claims.
Retain full evidence: Keep eight answers, four intervals, tie reasoning, independent checks and actual later work.

Learning Check

Eight rounding results: Name accuracy and avoid leading-zero counting.
Four bound intervals: Use the stated positive half-up convention.
Why exclude the upper endpoint: Check actual tie outcomes.
Estimation and independent checks: Distinguish scale checking from final precision.
New mass interval: Show units and both endpoints.
First errors and scope: Keep complete work and accurate interpretation.

Worked checks and assessment guidance

Results 3.5, 3.46, .0049, .005, 7900, 8000, 10.0, 12.35 with original units. Row 7 has a carry while preserving fixed tenths rounding; explicitly state nearest.1 rather than infer a different unit from significant-figure display.
17.5≤x<18.5 cm; 3.45≤x<3.55 kg; 115≤x<125 s; .00595≤x<.00605 m, with each unit applying to both endpoints. The half-width is half the stated rounding increment.
17.5 rounds to 18 while 18.5 rounds to 19, so the source upper endpoint is excluded. Values arbitrarily close below 18.5 can still round to 18; the upper bound is not an attained maximum.
19.8×4.12≈20×4=80 versus exact 81.576.48.6÷2.05≈50÷2=25 versus about 23.7073. Endpoint midpoint 18/width 1 and midpoint.0060/width.0001 reconstruct rounding data.
2.40 kg to nearest.01 kg gives 2.395≤m<2.405 kg, or 2395≤m<2405 g. Width.01 kg equals 10 g; the lower tie rounds to 2.40 and upper to 2.41.
Leading zeros do not begin significant figures; use half rather than a whole rounding unit on each side, and do not include an upper tie that rounds upward. These rounding bounds do not include every real measurement error or certify practical instrument competence.

Learning evidence

Retain eight rounding decisions, four fully signed unit-aware intervals, endpoint/tie reasoning, two estimates, inverse centre-width checks, first-error correction and new-unit transfer.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, Source area: Key stage 3 > Subject content > Number. Fluency, reasoning and problem solving. Year 9 allocation is editorial within KS3. Numerical models, tasks and feedback are original; images retain actual Commons attribution and videos name their original publisher. Reviewed 12 November 2026.

Decimal Place Value Chart (emphasizing the One's Column).jpg, GregWelch8, CC0. Unchanged original; caption and learning activities are original.

Dienes Material (Dyskalkulie).jpg, Rhetos, CC0. Unchanged original; caption and learning activities are original.