Estimation (rounding)

1Summary 重點摘要

Use rounding to get approximate values quickly.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The rounding ruleTo round 38 to the nearest ten, look at the ones digit: it is 8, and 8 is 5 or more, so round up and 38 ≈ 40. The motto reads: a tail of 5 or more rounds up, below 5 rounds down; that is rounding. Why is 5 the boundary? Because 38 differs from 30 by 8 but from 40 by only 2, so 40 is the nearest multiple of ten, and estimation must take the nearest value to be meaningful. A common error is dropping the ones digit without looking, writing 38 ≈ 30; check the tail first, then decide to round up or down, never the other way round.
2. Step 2: Rounding downThe ones digit of 23 is 3, and 3 is below 5, so it is dropped and 23 ≈ 20. After rounding down, the tens digit stays unchanged while the ones digit becomes 0. Why can it simply be dropped? Because 23 differs from 20 by only 3 but from 30 by 7, making 20 the nearest ten. Rounding down leaves the estimate a little smaller than the number, and rounding up a little larger; both are normal, since an estimate is by nature a near value. A common mistake is rounding up even here, writing 23 ≈ 30; keep the motto in mind — round up only at 5, and 4 or below always rounds down.
3. Step 3: Carrying in a chainFor 99 the ones digit is 9, so it rounds up; after adding one, the tens 9 becomes 10 and carries again, giving 100. Handle a chain of carries one place at a time, never skipping. Why does 99 become a three-digit number? Because 99 differs from 100 by only 1 but from 90 by 9, so the nearest ten naturally crosses 100. Similarly 98 ≈ 100 and 95 ≈ 100. A common error is losing control of the chain and writing 90, or missing the second carry; numbers near a hundred are especially slippery, so afterwards verify with the distance to 100.
4. Step 4: To the nearest hundredRounding to the nearest hundred looks at the tens digit: 462 has 6 in the tens place, and 6 is 5 or more, so it rounds up to 500; 340 has 4 there, below 5, so it rounds down to 300. The rule is identical to rounding to ten; only the digit being looked at moves from the ones to the tens. Why the tens digit? Because the decision about the hundreds place must come from the adjacent lower place — the universal principle of rounding: whichever place you aim for, look at the digit right after it. A common error is looking at the ones instead, writing 462 ≈ 400; confirm the target place first, then read the adjacent tail.
5. Step 5: Estimates check answersAn estimate can test a calculation: 38 × 22 is about 40 × 20 = 800, and the exact answer 836 is close, so the working looks sound. Why does this work? Because rounding errors are limited, and the exact answer should land near the estimate; if the exact answer comes out over eight thousand, ten times the estimate of 800, something has gone wrong and must be rechecked. This is an excellent self-check in tests, catching big mistakes within seconds. A common error is estimating and never comparing, or choosing poor approximations such as turning 22 into 30, which spoils the check; estimate sensibly and compare seriously.
6. Step 6: Use the approximation signAn estimate is not equal to the original number, so the approximation sign is required: write 38 ≈ 40, never 38 = 40. The sign reads as approximately equal to, meaning close to that number with a small gap. Why keep the two signs apart? Because the equals sign declares exact equality, and 38 and 40 are plainly unequal; writing = is a mistake of mathematical language and misleads the reader. For example 99 ≈ 100 is correct, while 99 = 100 is wrong. A common error is writing = everywhere for convenience, or mixing the signs at random; reserve the approximation sign for every estimated answer, and keep = for exact statements.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Round up on ≥ 5, down on < 5 (look at the next digit)
• 38 ≈ 40 and 23 ≈ 20
• 99 chains up to 100
• Estimates use ≈, never =
• Estimating quickly checks if an answer is sensible