KS3 rounding, estimation and error intervals: a parent’s guide

Rounding, estimation and error, explained

The topic starts with measures. Children are expected to be comfortable with standard units of mass, length, time and money, including decimal quantities such as 2.35 kg, 0.6 m or £4.50. The classic trap is time: 2.5 hours is 2 hours 30 minutes, not 2 hours 50, because a decimal hour is a fraction of 60 minutes rather than of 100. Money too: £12.5 on a calculator display must be written £12.50.

Rounding is taught in two ways. Rounding to a number of decimal places counts digits after the decimal point, so 3.14159 is 3.14 to 2 decimal places. Rounding to significant figures counts from the first digit that isn’t zero, so 48,653 is 49,000 to 2 significant figures and 0.004872 is 0.0049 to 2 significant figures. The skill that matters is choosing an appropriate degree of accuracy: a builder quoting a beam as 3.2 m doesn’t need 3.2047 m, and a minibus problem needs rounding up, because 53 people in 16-seat buses gives 3.3125 and you cannot hire a third of a bus.

Estimation means rounding every number in a calculation, usually to one significant figure, so the calculation becomes easy enough to do in your head. For 48.7 × 3.1 a child writes 50 × 3 = 150, and the exact answer of 150.97 is reassuringly close. For 412 ÷ 0.52 they might use 400 ÷ 0.5 = 800, and the exact answer is about 792. The wavy equals sign ≈ means “is approximately equal to”. An estimate is not a guess: it is a quick calculation that should land in the right neighbourhood, and it catches slips such as a misplaced decimal point.

Rounding means information is lost, and Key Stage 3 asks children to describe exactly how much. If a length is 7 cm to the nearest centimetre, the true length could be anything from 6.5 cm up to, but not including, 7.5 cm. This is written with inequality signs as 6.5 ≤ x < 7.5 and is called an error interval. The lower end uses ≤ because 6.5 itself rounds up to 7, and the upper end uses < because 7.5 would round up to 8. Children learn that the interval stretches half a rounding unit either side of the rounded value, and they use it to work out how far an answer calculated from rounded measurements could be out. Working with upper and lower bounds in harder, multi-step calculations comes later, at GCSE.

Finally, children use a calculator and other technology, such as a spreadsheet, to get accurate results and then interpret them. A calculator does exactly what it is typed, so 24 ÷ 4 + 2 is not the same as 24 ÷ (4 + 2). The display needs interpreting too: 3.3125 on the screen means four buses, a total of 12.5 means £12.50, and an answer that disagrees with the estimate is a signal to retype it rather than to trust the machine.

Words your child will hear

Decimal place

A digit to the right of the decimal point. 3.14 has two decimal places.

Significant figure

A digit that carries meaning, counted from the first non-zero digit. 0.0049 has two.

Degree of accuracy

How precisely an answer is given, such as to 1 decimal place or to 3 significant figures.

Estimate

A quick approximate answer found by rounding the numbers first.

≈

The symbol for “is approximately equal to”.

Inequality

A statement using <, >, ≤ or ≥, such as 6.5 ≤ x < 7.5.

Error interval

The range of values a rounded number could really have been before it was rounded.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

• Use standard units of mass, length, time, money and other measures, including with decimal quantities
• 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
• Use a calculator and other technologies to calculate results accurately and then interpret them appropriately

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Common mistakes and how to help

How to help at home

• Estimate before you check. At the shops, ask your teenager to estimate the total of the trolley to the nearest £10, or the cost of 7 items at £2.89, before the till gives the answer. Rounding to one significant figure becomes second nature.
• Ask “how accurate does it need to be?”. A recipe, a journey time and a bank balance need different accuracy. Talk through when to round up (carpet, paint, taxis), when to round down (how many portions) and when to keep the pence.
• Round a long number two ways. Pick a long decimal from a calculator, such as 7 ÷ 9, and ask for it to 2 decimal places and then 3 significant figures. Then try 0.00734 to see why the two methods differ.
• Play error-interval detective. Measure something with a ruler to the nearest centimetre, say 24 cm, and write the range it could be: 23.5 ≤ x < 24.5. Then ask how much a rectangle’s area could vary using the smallest and largest possible sides.
• Check decimal times. Ask what 1.25 hours is in hours and minutes (1 hour 15 minutes) and what 2 hours 20 minutes is as a mixed number of hours. It is a quick way to confirm they have not slipped into thinking in hundreds.

Related topics

Everything in Key Stage 3 maths