Supply Drive Packing: Remainders, Checks, and Reasonable Estimates
The fourth-grade team at Pine Ridge Elementary has 1,248 pencils and 875 erasers donated for a supply drive. Pencils go into boxes of 24; erasers go into bags of 15. The principal's rule is that no bag or box may be partly empty when the truck arrives at 1:00, so the team must decide what to do about leftovers. Three students produce conflicting results. Ana writes 1,248 ÷ 24 = 52 and says the pencil job is finished. Ben writes 875 ÷ 15 = 58 and says 5 erasers are “lost,” then checks by multiplying 58 × 15, gets 870, and still calls his answer correct. Cara checked Ana with 1,200 ÷ 20 = 60 and now doubts the quotient 52. The teacher has 25 minutes to decide which results to accept, which misconception to reteach to the whole class, and what to tell the parent volunteer who is counting bags.
1. Ana claims the pencil job is finished. Evaluate 1,248 ÷ 24 and explain what the quotient and remainder mean in this situation.
Read the worked answer
Use partial quotients: 24 × 50 = 1,200, and 1,248 − 1,200 = 48; then 24 × 2 = 48, so 50 + 2 = 52 with remainder 0. Check by multiplying: 52 × 24 = (50 × 24) + (2 × 24) = 1,200 + 48 = 1,248. A remainder of 0 means every pencil lands in a full box and nothing is left on the table, so Ana's statement is accurate - for pencils only. The teacher can sign off on the pencils but must not transfer that conclusion to the erasers, where the nonzero remainder forces a decision about the principal's rule.
2. Ben claims 875 ÷ 15 = 58 with 5 left over, and says his multiplication check proves he is right. Explain what the check actually shows and decide how many bags the team should prepare.
Read the worked answer
The check shows 58 × 15 = 870, which is 5 short of 875 - so the check confirms the remainder rather than erasing it. A correct check has two parts: quotient × divisor + remainder = dividend (58 × 15 + 5 = 875) and remainder < divisor (5 < 15). Because 58 bags hold only 870 erasers, the team must either prepare 59 bags (the last one holding 5) or gather 10 more erasers so the 59th bag is full. Under the principal's no-partly-empty rule, the honest choice is to set the 5 aside and label them, or add 10. Ben needs the two-part check modeled, not just corrected.
3. Cara estimated 1,200 ÷ 20 = 60 and now doubts that 52 can be right. Is her estimate a valid check? Explain and give a better estimate.
Read the worked answer
Cara changed both numbers, and she changed the divisor in the direction that inflates a quotient: dividing a smaller dividend by a much smaller divisor pushes the answer up. Good compatible numbers keep the divisor close to 24, for example 1,200 ÷ 24 = 50, or 2,400 ÷ 24 = 100 halved to 50. Since 52 is only 2 away from 50, the exact answer is reasonable; Cara's 60 is about 15% high, which is a sign her estimate is the weak link, not Ana's quotient. The reteach point: an estimate checks an answer only when the numbers used stay close to the original numbers.