Rounding to the Nearest 10, 100, 1000
Questions designed to stretch thinking, reveal misconceptions, and spark mathematical reasoning.
Convince Me That…
Students must construct a mathematical argument for why each statement is true.
450 sits exactly halfway between 400 and 500. On a number line, it is 50 away from 400 and 50 away from 500. When a number is exactly halfway (equidistant), the convention is to round up.
To round 347 to the nearest 10, we look at the units digit: 7. Since 7 ≥ 5, we round up to 350. To round 347 to the nearest 100, we look at the tens digit: 4. Since 4 < 5, we round down to 300.
The key insight is that different rounding levels use different deciding digits. The units digit (7) decides for nearest 10; the tens digit (4) decides for nearest 100. One tells us to round up, the other to round down — giving different answers of 350 and 300.
343 rounded to the nearest 10 is 340, which is smaller than 343. Whenever the deciding digit is 0, 1, 2, 3, or 4, we round down, and the rounded answer is less than or equal to the original number.
Rounding can make a number bigger (347 → 350), smaller (343 → 340), or leave it unchanged (340 → 340). It goes to the nearest multiple — and “nearest” can be above or below.
995 sits between 990 and 1,000 (the two nearest multiples of 10). The units digit is 5, so we round up — giving 1,000.
Some students resist this because 1,000 has four digits while 995 has three. But rounding doesn’t care about the number of digits — it simply finds the nearest multiple. 995 is equidistant from 990 and 1,000 (both 5 away), and since it’s exactly halfway, the convention is to round up — giving 1,000. The answer can absolutely cross into a new place-value range.
Give an Example Of…
For each prompt, provide: an example, another example, one no-one else will think of, and one someone might think works but doesn’t.
Example: 463
Another: 537
Creative: 500 itself — it’s already a multiple of 100, so it rounds to itself. Or 450 — exactly on the boundary, rounds up by convention.
Trap: 448 — a student might think “448 is close to 500” but the tens digit is 4 (< 5), so it rounds down to 400, not up to 500.
Example: 203 — rounds to 200 (nearest 10) and 200 (nearest 100)
Another: 504 — rounds to 500 (nearest 10) and 500 (nearest 100)
Creative: 1,000 — any multiple of 100 is already a multiple of both 10 and 100, so it rounds to itself at both levels.
Trap: 350 — rounds to 350 (nearest 10) but 400 (nearest 100). The tens digit of 5 causes rounding up to the next hundred, giving different answers at the two levels.
Example: 4,567 → 4,570 (nearest 10), 4,600 (nearest 100), 5,000 (nearest 1,000)
Another: 1,234 → 1,230 (nearest 10), 1,200 (nearest 100), 1,000 (nearest 1,000)
Creative: 5,555 → 5,560, 5,600, 6,000 — every deciding digit is 5, so every level rounds up!
Trap: 3,000 → rounds to 3,000 at all three levels. Any multiple of 1,000 is already a multiple of 10 and 100 too, so the three answers are identical, not different.
Example: 445 → nearest 10 gives 450 → nearest 100 gives 500. But 445 directly to nearest 100: tens digit is 4, so it rounds to 400. Two-step gives 500; one-step gives 400.
Another: 245 → 250 → 300 (two-step), but 245 → 200 (direct). Same pattern.
Creative: 1,945 → 1,950 → 2,000 (two-step), but 1,945 → 1,900 (direct). Here the cascading error crosses a thousands boundary!
Trap: 460 → nearest 10 is 460 → nearest 100 is 500. Direct to nearest 100 is also 500. The two methods agree. This cascading rounding error only happens when the tens digit is 4 but the units digit is 5 or more — so that rounding to the nearest 10 pushes the tens digit up to 5.
Always, Sometimes, Never
Is the statement always true, sometimes true, or never true? Students should justify their decision with examples.
Rounding to the nearest 10 means finding the nearest multiple of 10. Every multiple of 10 ends in zero: 10, 20, 30, 40, … 100, 110, and so on. So the result must always end in zero.
This is true whether the original number rounds up (37 → 40) or down (34 → 30). The rounded answer is always a multiple of 10, which always ends in 0.
It changes most numbers: 347 → 350, 382 → 380. But if the number is already a multiple of 10, rounding it to the nearest 10 leaves it unchanged: 340 → 340, 500 → 500.
This catches the assumption that rounding always does something. The units digit decides: if it’s 0, the number is already “rounded” and nothing changes. For any other units digit, the number will be adjusted.
Most three-digit numbers stay three digits: 347 → 300, 672 → 700. But 950 rounded to the nearest 100 is 1,000 — a four-digit number. Any three-digit number from 950 to 999 rounds up to 1,000, crossing the boundary.
Students are often surprised that rounding can change the number of digits. This links to the broader idea that rounding finds the nearest multiple — and that multiple might be in a different “range” than the original number.
The deciding digit is always one column to the right of the column you’re rounding to. To round to the nearest 100, you look at the tens digit. To round to the nearest 10, you look at the units digit. To round to the nearest 1,000, you look at the hundreds digit.
Think of it this way: The column you are rounding to is the Driver (it gets us where we are going). The column to its right is the Navigator (it tells the driver which way to turn).
Odd One Out
Which is the odd one out? The challenge: make a valid mathematical case for each one being the odd one out.
Explain the Mistake
Each example contains a deliberate error targeting a common misconception. Can you find where and why the reasoning goes wrong?
Answer: 300
Reasoning: “First I round 245 to the nearest 10, which is 250. Then I round 250 to the nearest 100. The tens digit is 5, so I round up to 300.”
The student has used cascading (two-step) rounding — rounding to the nearest 10 first, then rounding that result to the nearest 100. This two-step process can change the answer because the first round pushes the tens digit from 4 up to 5.
The correct approach is to round directly: look at the original tens digit of 245, which is 4. Since 4 < 5, round down. 245 rounds to 200, not 300. You should always round in one step from the original number.
Answer: 680 ✔
Reasoning: “I look at the tens digit. It’s 7, and 7 is bigger than 5, so I round up to 680.”
This is a False Positive. The student got the correct answer (680) but for the wrong reason. They looked at the tens digit (7) instead of the units digit (8). They only got it right because the units digit (8) also happened to round up.
This is dangerous because they will get 50% of questions right by accident, hiding the misconception. A counterexample breaks this immediately: try 738. The student’s method says “tens digit is 3, which is less than 5, so round down to 730.” But the correct answer is 740.
Answer: 3,000
Reasoning: “I found the hundreds digit, which is 4. Since 4 is less than 5, I round down. The answer is 3,000.”
The student has confused rounding to the nearest 100 with rounding to the nearest 1,000. They correctly identified the hundreds digit (4) but then rounded to the nearest 1,000 instead of the nearest 100. The hundreds digit doesn’t decide for rounding to the nearest 100 — it decides for rounding to the nearest 1,000.
To round 3,462 to the nearest 100: look at the tens digit, which is 6. Since 6 ≥ 5, round up. The correct answer is 3,500. The student used the right digit for the wrong rounding level.
Answer: 400
Reasoning: “To round to the nearest 100, I just remove the tens and units digits and replace them with zeros. So 467 becomes 400.”
The student is treating rounding as truncation — simply chopping off the unwanted digits. This ignores which multiple the number is actually closer to.
The correct method: look at the tens digit, which is 6. Since 6 ≥ 5, round up. 467 rounds to 500, not 400. Truncation is particularly dangerous because it gives the right answer about half the time (whenever the number genuinely rounds down).