The Food Drive Numbers
The third grade is packing bags for a food drive. There are 4 classes. Each class fills 6 boxes. Each box holds 9 cans. The gym where the boxes are stacked is 12 meters long. A sign says the school goal is 200 cans. Ravi solves: '4 × 6 = 24 boxes; 24 × 9 = 216 cans. We beat the goal by 16.' Tasha says, 'You forgot the 12 meters.' Ravi says, 'Meters are not cans.' Their teacher, Ms. Cole, asks the class three things: whether 216 answers the question, whether the 12 meters belongs in the work, and whether 216 is reasonable. Jayden says, 'I rounded first: 4 × 6 = 24, and 24 × 9 is about 24 × 10 = 240, then take away one 24 to get 216. That is close to 200, so it makes sense.' Ms. Cole asks the class to find a question the 12 meters would answer.
1. Should the 12 meters be used in this problem? Explain what goes wrong if someone multiplies by it.
Read the worked answer
No, the 12 meters should not be used.
- Units check: the question asks for cans. Classes × boxes × cans per box gives cans. Multiplying by meters would give 'can-meters,' which is not a thing anyone can count.
- Size check: 4 × 6 × 9 × 12 = 2,592 cans. That is more than twelve times the goal, for a stack of boxes in one gym. An answer that far off is a signal that an extra number got used.
- Meaning check: 12 meters describes the gym floor, not the food. Nothing about how long the room is changes how many cans are in a box.
Word problems often include a number that is there to describe the setting. The judgment is to ask of every number: does the question ask for this kind of quantity? If not, leave it out and say why.
2. Is 216 correct and reasonable? Show two different checks.
Read the worked answer
216 is correct, and it beats the 200 goal by 16.
Check 1 - break apart: 24 × 9 = (24 × 10) − 24 = 240 − 24 = 216.
Check 2 - split the 24: (20 × 9) + (4 × 9) = 180 + 36 = 216.
Check 3 - go backwards: 216 ÷ 9 = 24 boxes, and 24 ÷ 6 = 4 classes. That matches the starting facts.
Reasonableness: 24 × 9 should be a little under 24 × 10 = 240, and 216 is a little under 240. It is also close to the goal of 200, which fits the story - a school that set a goal of 200 and filled 24 boxes would land near it. An answer like 2,592 or 2160 would fail this check instantly.
3. Write one question that needs the 12 meters, and one question that does not need the 9.
Read the worked answer
A question that needs the 12 meters:
'How many meters of tape are needed to mark the length of the gym floor where the boxes are stacked?'
Here the answer is a length, so meters are the right unit. (If the question asked for tape around the whole rectangle, the width would also be needed - a good reminder that a number is only usable when the question asks for that quantity.)
A question that does not need the 9:
'How many boxes will the four classes fill in all?'
That is 4 × 6 = 24 boxes. The cans per box never come up, because nothing in the question is counted in cans.
The skill being tested: decide which numbers the question actually calls for, and be able to say what a leftover number would be good for.