Rounding Before Adding: Which Estimate Is Defensible?
Ms. Okafor's class is working on estimation. The problem on the board reads: 'The school store had 348 pencils on Monday and 249 pencils on Tuesday. About how many pencils is that altogether?' Jada rounds both numbers to the nearest hundred and writes 300 + 200 = 500. Marcus rounds both numbers to the nearest ten and writes 350 + 250 = 600. Priya ignores the word 'about' and computes 348 + 249 = 597. Devon computes 348 + 249 and writes 5,097. Later, the same class estimates 249 + 251, and several students say the answer is about 400 while others say 500. Ms. Okafor has to decide which estimates are defensible, which errors are about rounding and which are about place value, and how to respond to Priya without teaching the class that precision is wrong.
1. For the pencil problem, which estimates are defensible answers to 'about how many,' and which students actually answered the question asked? Use the exact sum to justify your judgment.
Read the worked answer
The exact sum is 348 + 249 = 597.
- Jada (500): Her rounding is correct. 348 rounds to 300 because 48 < 50; 249 rounds to 200 because 49 < 50. The estimate is 97 away from the exact sum. Defensible as an estimate - it is within about a hundred and it correctly rounds each addend to the nearest hundred, which is the grade-level expectation.
- Marcus (600): Also correct rounding, to the nearest ten (350 and 250). It is only 3 away from the exact sum, so it is the closer estimate. Defensible.
- Priya (597): Correct arithmetic, but she answered 'how many' rather than 'about how many.' She should not be told she is wrong; she should be asked which word in the question tells the reader an estimate is wanted, and then asked to produce an estimate and keep her exact answer as a check.
The judgment to teach: an estimate is judged by whether the rounding was done correctly and whether it is close enough to be useful, not by whether it matches the exact sum.
2. Devon wrote 5,097. Diagnose the error, distinguish it from a rounding error, and describe how you would confirm the diagnosis.
Read the worked answer
Devon's digits are the correct digits (5, 9, 7) in the correct order - he placed the leading 5 in the thousands place instead of the hundreds place. This is a place-value and alignment error, not a rounding error and not a regrouping error: no correct regrouping of 8 + 9 produces a thousands digit, so 5,097 cannot be reached by any legal carrying.
Confirm it, don't assume it. Ask him to read his answer aloud ('five thousand ninety-seven') and then say how many pencils that would be. Then have him build 348 and 249 with base-ten blocks on a place-value chart and combine them. If he writes 5 hundreds and then misplaces them on the chart, the chart is the issue; if he builds 597 correctly but writes 5,097, the issue is transcription or column alignment when the numbers are written in vertical form.
Use the estimate as the tool: 5,097 is nowhere near 500 or 600. Teach the habit of checking a sum against the estimate, which would have caught this before the paper left the desk.
3. For 249 + 251, explain how a student could reasonably arrive at 400, why 500 is the better estimate, and what representation would settle the disagreement.
Read the worked answer
Correct rounding to the nearest hundred: 249 → 200 (49 < 50) and 251 → 300 (51 > 50), giving 200 + 300 = 500. The exact sum is also 500, so the estimate happens to be exact.
A student reaching 400 has almost certainly rounded 251 down to 200. Two common causes: looking only at the hundreds digit ('both start with 2, so both are 200'), or misapplying the 'look at the next digit' rule and treating 251 as if the tens digit were 4 or less. A third possibility is treating 249 and 251 as 'close together, so round them the same way.'
The representation that settles it: a number line from 200 to 300 with 250 marked at the midpoint. Place 249 to the left of 250 and 251 to the right. The rule '5 or more rounds up' is a shortcut for 'which side of the midpoint are you on,' and the line makes that visible. Have students place 245, 249, 250, and 251 and describe each in relation to 250. Flag 250 explicitly - the rounding convention at the exact midpoint needs to be stated, not discovered.