Sharing Stickers at the Class Fair
A Grade 3 class has 68 stickers to give away at the fair. They want to put the same number of stickers into each of 5 goodie bags, and any stickers left over go back into the supply box. Maya says, "68 divided by 5 is 13 remainder 3, so each bag gets 13 stickers and 3 go back." Leo says, "No, 68 divided by 5 is 12 remainder 8, so each bag gets 12 and 8 go back." Their teacher asks them to check with multiplication and to think about whether a remainder bigger than the divisor can be correct. The class also wonders what would happen if they used 6 bags instead of 5, and whether the number going back to the box would be larger or smaller.
1. Use multiplication to decide whose division is correct. Explain what the remainder means in this situation and why a remainder of 8 cannot be right when dividing by 5.
Read the worked answer
Check Maya: 5 × 13 = 65, and 65 + 3 = 68. This matches, so Maya is correct.
Check Leo: 5 × 12 = 60, and 60 + 8 = 68. The multiplication also matches, but the remainder 8 is a problem. A remainder must be smaller than the divisor. If you had 8 left over, you could still put 1 more sticker into each of the 5 bags (that uses 5 of the 8), leaving only 3. So 8 is not a finished remainder.
With 5 bags: each bag gets 13 stickers, and 3 stickers go back to the box. The remainder 3 means 3 stickers cannot make another full round of 5 bags.
2. Now suppose the class uses 6 bags instead of 5, still sharing all 68 stickers equally. Find the quotient and remainder, and decide whether more or fewer stickers go back to the box than with 5 bags. Justify your answer.
Read the worked answer
Divide 68 by 6. Since 6 × 11 = 66 and 6 × 12 = 72, the quotient is 11. Then 68 − 66 = 2, so the remainder is 2.
Check: 6 × 11 + 2 = 66 + 2 = 68. Correct.
With 6 bags, each bag gets 11 stickers and 2 go back. With 5 bags, 3 went back. So with 6 bags, fewer stickers go back (2 instead of 3).
Reasoning: more bags means each bag holds fewer stickers, but the leftover depends on how close 68 is to a multiple of the number of bags. 68 is 2 away from 66 (a multiple of 6) but 3 away from 65 (a multiple of 5), so the 6-bag plan leaves a smaller remainder.