Interpreting a Remainder When Packing Pencils
A class is packing 138 pencils into boxes of 8 for a community event. The teacher asks, “How many boxes are needed?” Student M writes 138 ÷ 8 = 17 R 2 and answers 17 boxes. Student P answers 18 boxes. The teacher also has a budget of $45; each box costs $2.50, and only whole boxes can be purchased. The class must decide how many boxes to buy and whether any pencils will be left over. The teacher wants students to explain the meaning of the remainder in context and to check the answer against the budget. Several students say “the remainder is the leftover pencils, so it doesn’t matter.” The school requires every pencil to be in a box.
1. How many boxes are needed, and how should the remainder be interpreted in this situation?
Read the worked answer
18 boxes are needed. 138 ÷ 8 = 17 remainder 2 means 17 boxes hold 136 pencils, with 2 pencils left over. Because every pencil must be in a box, those 2 pencils require one more box. Student M’s answer of 17 boxes leaves 2 pencils unpacked. The remainder is not simply a number to report; it represents pencils that still need a box. This is a “round up” remainder situation, not a “drop the remainder” situation.
2. Check the cost against the $45 budget. Does the class have enough money? Show the reasoning.
Read the worked answer
18 boxes at $2.50 each cost 18 × 2.50 = 45 dollars, so the class has exactly enough money. A useful check is to compute 17 boxes first: 17 × 2.50 = 42.50, then add one more box: 42.50 + 2.50 = 45.00. Since the budget is $45, buying 18 boxes uses the entire budget and leaves no money for tax or extra supplies. If the class bought only 17 boxes, it would spend $42.50 but would not pack all 138 pencils.
3. A student says, “The remainder is 2, so we just write 17 R 2 and stop.” Write a corrected explanation that the student could use.
Read the worked answer
The division 138 ÷ 8 = 17 R 2 tells us that 17 full boxes hold 136 pencils and 2 pencils are left. The question asks how many boxes are needed to pack all the pencils. The 2 leftover pencils still need a box, so the answer is 18 boxes. The remainder changes the answer because the context requires every pencil to be packed. A complete explanation should include the multiplication check 17 × 8 = 136, add the remainder 136 + 2 = 138, and then state why one more box is necessary.