Bounds and error intervals
Writing the error interval for a rounded measurement, and finding upper and lower bounds in a calculation
About three lessons of 45 to 60 minutes
A luggage scale says 20 kg. Could the suitcase really weigh 20.4 kg?
Scales, food packets and journey estimates round numbers constantly: '20 kg', '250 g', 'takes 30 minutes'. But a rounded measurement is not the exact truth, it is the CENTRE of a whole range of possible true values. If a luggage scale reports 20 kg to the nearest kilogram, the actual mass could be anywhere from 19.5 kg up to (but not including) 20.5 kg, so 20.4 kg is entirely possible.
This matters whenever measurements feed into another calculation: an engineer checking tolerances, a carrier calculating total parcel mass, or a GCSE exam question asking for an upper or lower bound all rely on exactly this idea. This unit teaches you to turn any rounded measurement into its full error interval, and to carry that uncertainty correctly through a calculation.
- A suitcase measured as 20 kg, to the nearest kilogramits true mass could be anywhere from 19.5 up to (not including) 20.5 kg
- A sprinter's time of 12 seconds, to the nearest secondtheir true time is somewhere in 11.5 to 12.5 seconds
- A parcel's mass of 2 kg, to the nearest kgcould really be as light as 1.5 kg or as heavy as (just under) 2.5 kg
- A car's rounded distance and time on a dashboardcombining two rounded readings needs the bounds of BOTH to find a genuine max or min speed
What students will be able to do
Students will find the lower and upper bound of a value given its rounding accuracy, write the error interval using inequality notation, work backwards from an interval to the rounded value and its accuracy, and use bounds to find the maximum or minimum possible result of a sum, difference, product or quotient of rounded measurements.
- I can find the lower bound and upper bound of a value rounded to a stated accuracy (nearest whole number, nearest 10, nearest 100, ...).
- I can write an error interval using inequality notation, LOWER ≤ x < UPPER.
- I can work backwards from an error interval to find the original rounded value and its degree of accuracy.
- I can find the maximum and minimum possible value of a sum or product by choosing the correct combination of bounds, not just always the two upper or two lower bounds.
- I can find the maximum and minimum possible value of a quotient (like speed = distance / time), knowing the max needs the largest numerator with the smallest denominator.
Standards this unit teaches
- GCSE Number #15UK GCSE Mathematics (DfE, England)Error intervals from rounding
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Number" section, "Measures and accuracy", item 15, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "round numbers and measures to an appropriate degree of accuracy (...); use inequality notation to specify simple error intervals due to truncation or rounding". The rounding half is standard type; the error-interval half is underlined. Standard + underlined together make up the full Foundation tier (all GCSE students taught and assessed).
- GCSE Number #16UK GCSE Mathematics (DfE, England)Upper and lower bounds
Subject content statement (same document, item 16): students should "apply and interpret limits of accuracy" (underlined type, so Foundation tier), ", including upper and lower bounds" (bold type, so Higher tier only). The error-interval and single-measurement bounds work in this unit is Foundation; the bounds CALCULATIONS (max/min of a sum, product, quotient or difference of rounded quantities) taught in the later sections sit in the bold clause and are assessed on the Higher tier only.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Rounding unit
- the accuracy a value was rounded to, such as the nearest whole number, nearest 10, or nearest 100
- Lower bound
- the smallest value that would still round to the given rounded value
- Upper bound
- the smallest value that would round UP to the NEXT rounded value; the true value is always strictly less than this
- Error interval
- the full range of possible true values, written as lower bound ≤ x < upper bound
- Truncation
- cutting a number off after a certain digit without rounding (e.g. 7.89 truncated to 1 dp is 7.8, not 7.9)
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. From a rounded value to an error interval
ConcreteA value rounded to a given accuracy could have come from any true value within half of that accuracy either side. If a crowd of 350 people is reported to the nearest 10, the true count could be as low as 345 (which rounds down to 350) or as high as, but not quite reaching, 355 (which would round UP to 360 instead).
The rule: for a rounding unit u, lower bound = rounded value - u/2, and upper bound = rounded value + u/2. The upper bound uses a STRICT less-than, because a value exactly at the upper bound would actually round to the NEXT rounded value up, not this one.
A road sign gives a village's population as 350, correct to the nearest 10. Write the error interval for the true population, n.
- The rounding unit is 10, so half of it is 5.
- Lower bound = 350 - 5 = 345.
- Upper bound = 350 + 5 = 355.
Answer: 345 ≤ n < 355
- Why is the upper bound written with a STRICT less-than (<) but the lower bound with ≤?
- If a value is rounded to the nearest 100 instead of the nearest 10, how does the error interval change?
2. Working backwards: from an interval to the rounding
PictorialGiven only an error interval, you can recover BOTH the original rounded value and the accuracy it was rounded to. The rounded value always sits exactly in the middle of the interval, and the width of the interval always equals the rounding unit.
For 44.5 ≤ m < 45.5: the midpoint is (44.5 + 45.5) / 2 = 45, and the width is 45.5 - 44.5 = 1, so the original value was 45, rounded to the nearest whole number (kg, in a mass context).
A value x satisfies 995 ≤ x < 1005. What was the rounded value, and to what accuracy was it rounded?
- Midpoint = (995 + 1005) / 2 = 1000.
- Width = 1005 - 995 = 10.
- So the original value was 1000, rounded to the nearest 10.
Answer: 1000, correct to the nearest 10.
- If an interval has width 100, to what accuracy was the value rounded?
- Could two DIFFERENT rounded values ever produce the same error interval? Why not?
3. Upper and lower bounds in a calculation
AbstractWhen a calculation combines two rounded measurements, the maximum and minimum possible RESULTS do not both simply use 'the two upper bounds' or 'the two lower bounds': it depends on the operation. For a SUM or PRODUCT, the maximum comes from the two upper bounds and the minimum from the two lower bounds. For a QUOTIENT (division), the maximum comes from the LARGEST numerator divided by the SMALLEST denominator, and the minimum from the smallest numerator divided by the largest denominator.
Example (product, area): a rectangle is 8 cm by 5 cm, each to the nearest cm. Bounds: length 7.5 to 8.5 cm, width 4.5 to 5.5 cm. Maximum area = 8.5 x 5.5 = 46.75 cm2. Minimum area = 7.5 x 4.5 = 33.75 cm2.
Example (quotient, speed): a runner covers 15 km, to the nearest km, in 75 minutes, to the nearest minute. To get the FASTEST possible speed, use the LONGEST possible distance (15.5 km) over the SHORTEST possible time (74.5 minutes); to get the SLOWEST possible speed, use the SHORTEST distance (14.5 km) over the LONGEST time (75.5 minutes).
A runner covers 15 km, correct to the nearest km, in 75 minutes, correct to the nearest minute. Find the maximum possible average speed, in km/h, to 1 decimal place.
- Distance bounds: 14.5 to 15.5 km. Time bounds: 74.5 to 75.5 minutes, i.e. 74.5/60 to 75.5/60 hours.
- For the MAXIMUM speed, use the LARGEST distance and the SMALLEST time: 15.5 / (74.5/60).
- 15.5 x 60 / 74.5 = 930 / 74.5 = 12.483... ≈ 12.5 km/h.
Answer: 12.5 km/h (to 1 decimal place). The minimum possible speed, by the same method with the bounds swapped, is 14.5 / (75.5/60) ≈ 11.5 km/h.
- Why does the maximum speed use the SMALLEST possible time, not the largest?
- For a difference (a - b), which bound of a and which bound of b gives the MAXIMUM result?
Common misconceptions and how to address them
MisconceptionThe upper bound should be included with ≤, the same as the lower bound.
Why it happens: Students copy the ≤ symbol from the lower bound without noticing the interval is not symmetric in how it is written.
How to address it: A value exactly at the 'upper bound' would actually round UP to the next value, not the one given, so it can never truly be reached. Always write the upper bound with a strict < (or explain out loud: 'up to, but not including').
MisconceptionFor every calculation, the maximum result always uses the two upper bounds.
Why it happens: This is true for a sum or product, so students over-generalise it to division and subtraction too.
How to address it: For a QUOTIENT or DIFFERENCE, the maximum mixes bounds: largest numerator with smallest denominator (quotient), or largest first term with smallest second term (difference). Always ask 'which combination genuinely gives the biggest result?' rather than applying a fixed rule.
MisconceptionHalving the rounding unit means dividing by 2 in general, so nearest 10 gives a half-width of 10.
Why it happens: Students forget the crucial extra step: the error interval's half-width is HALF of the rounding unit, not the rounding unit itself.
How to address it: Always state the rounding unit first, then explicitly halve it: nearest 10 -> half-width 5; nearest 100 -> half-width 50; nearest whole number -> half-width 0.5.
Guided practice (with answers)
1. A bag of sugar has a mass of 2 kg, correct to the nearest kg. Write the error interval.
Answer: 1.5 ≤ m < 2.5, because half of 1 kg is 0.5 kg.
2. A sprint time is 12 seconds, correct to the nearest second. Find the upper bound.
Answer: 12.5 seconds, because 12 + 0.5 = 12.5.
3. A value x satisfies 235 ≤ x < 245. What was the rounded value and its accuracy?
Answer: 240, correct to the nearest 10, because the midpoint of 235 and 245 is 240 and the interval width is 10.
4. A rectangle is 6 cm by 4 cm, each correct to the nearest cm. Find the maximum possible perimeter.
Answer: 22 cm, because the bounds are 5.5 to 6.5 cm and 3.5 to 4.5 cm, so maximum perimeter = 2 x (6.5 + 4.5) = 22 cm.
5. A mass is given as 45 kg, correct to the nearest kg. Write the error interval.
Answer: 44.5 ≤ m < 45.5, because half of 1 kg is 0.5 kg.
6. Two lengths are 12 cm and 7 cm, each correct to the nearest cm. Find the minimum possible area of a rectangle with these dimensions.
Answer: 74.75 cm2, because the lower bounds are 11.5 cm and 6.5 cm, and 11.5 x 6.5 = 74.75.
Independent practice worksheets
Practise finding error intervals and bounds, and applying bounds to sums, products, differences and quotients, with computed, never-wrong answer keys.
Differentiation
- Always write the rounding unit and its half explicitly before touching the number, e.g. 'unit = 10, half = 5', so the halving step is never skipped.
- Use a number line (like the figure above) for every early example, marking the rounded value in the centre and the two bounds either side.
- Stick to whole-number rounding units (nearest 1, 10, 100) until the bound-finding process is automatic, before introducing decimal accuracies.
- For calculations, write out the FOUR bounds (both quantities' lower and upper) in a small table before deciding which pair to combine.
- Introduce decimal rounding accuracies (e.g. to the nearest 0.1) and confirm the half-width halving rule still applies.
- Ask students to justify, in words, why the maximum of a quotient needs a large numerator AND a small denominator, not just one or the other.
- Explore what happens to the width of an error interval as the rounding gets more precise (nearest 100 -> nearest 10 -> nearest 1): does the range of uncertainty grow or shrink?
- Pose a three-quantity calculation (e.g. volume from three rounded dimensions) and ask students to generalise the bound-combining rule themselves.
Assessment: exit ticket
A three-question exit ticket sampling error intervals, working backwards, and bounds in a calculation.
1. A distance is given as 240 m, correct to the nearest 10 m. Write the error interval.
Answer: 235 ≤ d < 245, because half of 10 is 5.
2. A value x satisfies 1150 ≤ x < 1250. What was the rounded value and its accuracy?
Answer: 1200, correct to the nearest 100, because the midpoint of 1150 and 1250 is 1200 and the interval width is 100.
3. Two masses are 12 kg and 7 kg, each correct to the nearest kg. Find the maximum possible total mass.
Answer: 20 kg, because the upper bounds are 12.5 kg and 7.5 kg, and 12.5 + 7.5 = 20.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 error intervals from rounding (section 1), Lesson 2 working backwards (section 2), Lesson 3 bounds in calculations plus the exit ticket (section 3 and assessment).
- This unit assumes comfort with rounding to a stated accuracy. Revisit the Grade 7 Rounding & Estimation worksheet first if the rounding step itself, not just the bound, is shaky.
- Curriculum note: DfE GCSE Number item 15 and item 16's 'apply and interpret limits of accuracy' clause are Foundation tier (standard/underlined), so error intervals and single-measurement bounds are for every GCSE student. Item 16's ', including upper and lower bounds' clause is BOLD, so the bounds-in-calculations work in Lesson 3 (max/min of a sum, product, difference or quotient of rounded quantities) is Higher tier only - treat it as extension material for a Foundation class.
- Language to repeat: 'half the rounding unit' for the bound gap, and for calculations, always ask out loud which specific combination of bounds gives the biggest or smallest possible answer, rather than a memorised shortcut.
- The number-line figure in section 1 is worth projecting and relabelling live with a different rounded value and unit, so students see the same structure (rounded value in the centre, bounds either side) apply generally.