Plan multistep solutions and interpret the remainder

Learning goal: Solve multistep whole-number problems using the four operations, represent unknowns with letters and interpret a division remainder in context. A remainder is the amount left after making as many complete equal groups as possible. What to do with it depends on the question, not on a universal instruction to round every answer the same way.

A class prepares art kits. There are seventy-three pencils, and each complete kit needs eight pencils. Nine complete groups use 9 × 8 = 72 pencils, leaving one. The division result is nine remainder one. If asked for complete kits, answer nine kits with one pencil left. A tenth complete kit cannot be made from the available pencils.

Now use the same quantities for a different situation: seventy-three learners need seats, and each vehicle has eight available learner seats in this mathematical model. Nine vehicles provide seventy-two seats, leaving one learner without a seat. Ten vehicles are needed. The remainder indicates a need for another vehicle, even though it will not be full. Real transport planning has other requirements, but this problem isolates the capacity relationship.

A remainder may itself be the requested answer. If seventy-three beads are packed into bags of eight and the question asks how many beads remain unpacked, answer one bead. Reporting nine bags would answer a different question. State what the quotient counts and what the remainder counts before choosing the final response.

Multistep problems combine these relationships with other actions. A club buys six boxes of twelve pencils and already has seventeen. It saves nine pencils for later and puts the rest into kits of eight. First calculate 6 × 12 = 72 pencils. Then add 17 to get 89. Subtract 9 to get 80 available for kits. Finally 80 ÷ 8 = 10 complete kits.

Represent the final unknown with a letter: ((6 × 12) + 17 − 9) ÷ 8 = k, so k = 10. Linked equations are equally useful when each intermediate result is labeled. The expression should follow the story's relationships; parentheses help make the intended groups clear. Do not stop after finding eighty when the question asks for kits.

Estimate to check reasonableness. In the kit problem, seventy-two plus about twenty is about ninety. Saving about ten leaves about eighty, enough for about ten groups of eight. The exact result fits this estimate. Estimation does not replace exact arithmetic, especially near a capacity boundary where one extra item can require an additional container.

Check by rebuilding the original amount. For seventy-three divided by eight, 8 × 9 + 1 = 73. A valid whole-number remainder must be smaller than the divisor. If the remainder were nine, another group of eight could still be made, so the grouping would not be finished.

Worked multistep example: Four boxes contain twenty-five cards each. Thirteen cards are reserved, and the rest are placed in packs of six. The starting total is 100; after reserving thirteen, 87 remain. Fourteen full packs use 84, leaving 3 cards. If asked for full packs, answer fourteen. If asked for containers to hold all cards with at most six per container, fifteen are needed.

Worked error check: A learner reports thirteen remainder nine for 87 ÷ 6. Although 6 × 13 + 9 equals 87, the remainder is too large. Move six of the remaining nine into another pack to obtain fourteen remainder three.

Guided practice: Solve paired stories with the same arithmetic but different questions about full groups, leftover items or total containers. Model the remainder with objects or a labeled bar. Then solve a multistep problem and annotate each intermediate quantity.

Independent evidence: Present a plan, equations with an unknown, exact calculation, interpreted remainder and a reasonableness check. Explain why two situations with the same division can have different final whole-number answers. If the final number has no unit, identify what it counts before accepting the solution.