Division with remainders

Activities to try

1. 1

   Multi-step problem workbench: plan, carry, interpret

   Why it works: A multi-step problem is solved by planning a sequence of operations and carrying the output of one step into the next; the equals sign expresses relationships, and an up-front estimate plus an inverse check catch magnitude and step errors.

   1. Read a genuinely two-step problem where step 1's result feeds step 2 — e.g. 'A trip needs 52 children transported in minibuses holding 9; each minibus costs £20. What is the total cost?' (first find the number of minibuses, then multiply by the cost). Note: the pencil problem 'packs of 6 pencils for £1.50 each — how much do 5 packs cost?' is solved in a single step (5 × £1.50 = £7.50; the pack size 6 is not needed), so use it for operation-choice practice, not for carrying an intermediate result.
   2. Lay out a bar model for step 1, record the intermediate result, then build a second bar model for step 2 on top of that result rather than restarting.
   3. Write an equation that matches the plan, using a letter such as n for the unknown.
   4. Before computing a final figure, state an estimate of what a reasonable answer looks like.
   5. Compute, then check: does the answer match your estimate, and does the inverse operation confirm it?
2. 2

   Deciding the remainder: contrasting contexts

   Why it works: The same division can have opposite remainder treatments depending on the situation; you must interpret the leftover against what the context demands rather than apply a fixed rule or leave a bare remainder.

   1. Solve '52 children go on a trip in minibuses that hold 9 each — how many minibuses are needed?' Compute 52÷9 = 5 remainder 7.
   2. Decide what the leftover 7 means here: the 7 children still need transport, so round UP to 6 minibuses.
   3. Now solve the contrasting problem 'How many teams of 9 can you make from 52 children?' with the same arithmetic 52÷9 = 5 remainder 7.
   4. Decide what the leftover means here: 7 children cannot form a full team, so the answer is 5 teams (drop the remainder).
   5. State a sensible real-world answer each time — not 'r7' — and explain why the identical division is rounded differently; consider whether '5 and a bit' minibuses could ever make sense, and why not.
3. 3

   Check and match: operation choice, equation and estimate

   Why it works: Operation choice, equation-matching, remainder interpretation and reasonableness are all things you can check deliberately, and judging reasonableness is as much part of problem solving as computing.

   1. Given a short story and candidate remainder interpretations, choose the one that matches the context and say why.
   2. Match a story to the two-step equation (with a letter for the unknown) that correctly represents it, rejecting equations that mis-order the steps.
   3. Choose or state the best estimate to check a computed answer, and explain whether a given answer is reasonable.
   4. Where a remainder can sensibly be expressed as a fraction or decimal (e.g. sharing remaining items equally), decide whether round up, drop, keep, or fraction fits the story.