Rounding and Estimating

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infoWhy this? Rounding, truncation, and estimation develop judgement about precision and the usefulness of approximate values. They help us check calculations, communicate measurements appropriately, and recognise that numerical representations can contain uncertainty.

scheduleWhy now? After extensive calculation and percentage work, we are ready to distinguish exact values from approximations and state accuracy consistently. These place-value foundations lead to significant figures, bounds, and error intervals in later years.

Rounding is replacing a number with a nearby value that is easier to use, while keeping it as close as possible to the original number.
When rounding to the nearest 10, 100, or 1000, you keep the digit in the tens, hundreds, or thousands place and change all digits to the right to zero.
To round to the nearest whole number, you look at the first digit after the decimal point (the tenths digit).
When rounding, if the digit after the rounding place is 0–4, the rounding digit stays the same and the digits after it are replaced with zeros (or removed for decimals).
When rounding, if the digit after the rounding place is 5–9, the rounding digit increases by 1 and the digits after it are replaced with zeros (or removed for decimals).
The “rounding digit” is the digit in the place value you are rounding to; the “check digit” is the next digit to the right that decides whether you round up or down.
Rounding to a given number of decimal places (dp) means keeping that many digits after the decimal point and using the next digit to decide whether to round up or down.
Rounding to `n` decimal places is the same as rounding to the nearest 0.1, 0.01, 0.001, etc., depending on `n`.
A number is exactly halfway between two rounding targets when the check digit is 5 followed by only zeros (or when the remaining part is exactly 0.5 of the next unit).
When rounding to the nearest whole number, all numbers from `x` up to but not including `x + 0.5` round to `x`, and all numbers from `x + 0.5` up to but not including `x + 1` round to `x + 1`, where `x` is a whole number.
When rounding whole numbers to the nearest 10, all whole numbers from `n` to `n + 4` round to `n`, and all whole numbers from `n + 5` to `n + 9` round to `n + 10`, where `n` is a multiple of 10.
Truncation means cutting a number off after a chosen digit without rounding, so the digits after the cut are simply removed.
Truncating to `n` decimal places keeps the first `n` digits after the decimal point and deletes the rest, even if the next digit is 5 or more.
For positive numbers, truncation always gives a value that is less than or equal to the original number.
Rounding can increase or decrease a number, but truncation never increases a positive number.
The more decimal places you round or truncate to, the closer the result is likely to be to the original number.
Rounding to a larger unit (like the nearest 1000 instead of the nearest 10) usually produces a less precise approximation.
Place value columns for whole numbers increase by powers of ten (ones, tens, hundreds, and thousands), and decimal places decrease by powers of ten (tenths, hundredths, and thousandths).

rocket_launchYou must be able to

Round a whole number to the nearest 10, 100, or 1000 by identifying the rounding digit, checking the next digit, and replacing digits to the right with zeros.
Round a decimal to the nearest whole number by checking the tenths digit and then either keeping the ones digit the same or increasing it by 1.
Round a decimal to any given number of decimal places by locating the required decimal place, using the next digit to decide whether to round up or down, and then deleting all later digits.
Carry out rounding accurately when rounding up causes regrouping, for example, 3.99 to 1 dp becomes 4.0, or 999 to the nearest 1000 becomes 1000.
Truncate a number to any given number of decimal places by keeping the required digits and deleting all remaining digits without changing the last kept digit.
Explain a rounding decision using place value language, for example, “I checked the hundredths digit, so I rounded the tenths up.”
Identify the two neighbouring multiples, for example, the two nearest multiples of 100, and use them to decide which target a number rounds to.
Write the rounded or truncated answer in a correct form, including any necessary trailing zeros to show the stated accuracy, for example, 2.50 to 2 dp.

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