Choose what a remainder means for the job

diagnostic. Group 53 counters in groups of six. Complete 53 = 6 × 8 + __. Enter the leftover count and explain why another complete group cannot be made.

Answer: 5

Teacher response: Eight groups use 48 counters and leave five. A ninth full group would need one more counter. A remainder is smaller than the group size.

model. A science-kit organiser must transport all 137 indivisible specimen cards. Each envelope holds at most eight cards. How many envelopes are needed, and how many cards are in the last envelope?

1. Underline all 137 cards and at most eight per envelope. Draw an envelope as a container with eight spaces. Because every card must travel, a partly filled final envelope is allowed and a fraction of an envelope is not an available container.
2. Build the division record: 8 × 17 = 136 and 137 − 136 = 1, so 137 = 8 × 17 + 1. Label 17 as full envelopes and 1 as a leftover card. The leftover has a different unit from the quotient.
3. Open one more envelope for that card. Check both sides of the decision: 17 envelopes have capacity 136, which is insufficient; 18 have capacity 144, which is sufficient. There are seven unused card spaces across the final envelope.
4. Change only the job: if the organiser may make complete eight-card sets and keep spare cards, the result is 17 complete sets and one spare card. Compare this with the fractional ribbon shares in the transfer. The arithmetic record stays 17 remainder 1; the requested unit and allowed actions determine the decision.

Worked example: 18 envelopes are needed, with one card in the last envelope. Seventeen full envelopes hold 136 cards, so the remaining card needs an eighteenth envelope. The answer is a count of envelopes, not 17 remainder 1 envelopes.

p1. Support: all 29 indivisible cards must travel in envelopes holding at most eight each. Enter the number of envelopes. Draw the five cards left after three full envelopes.

Answer: 4

Teacher response: Three full envelopes hold 24 cards and leave five. One partly full envelope carries those five, giving four envelopes altogether.

p2. Core: there are 94 cards. A complete kit must contain exactly eight cards; incomplete kits are not issued and spare cards stay in storage. Enter the number of complete kits, and record the spare count on paper.

Answer: 11

Teacher response: Eleven kits use 88 cards and leave six. The job counts complete kits, so the six spare cards do not make a twelfth kit.

p3. Challenge: transport all 167 indivisible tiles in trays holding at most ten each, and take two additional empty spare trays. Enter the total number of trays taken and distinguish the partly filled tray from the empty spares.

Answer: 19

Teacher response: Sixteen full trays hold 160 tiles; one partly filled tray holds the last seven. Those 17 carrying trays plus two empty spares make 19.

e1. Independent check: all 146 indivisible cards must travel in envelopes holding at most nine each. Enter the minimum envelope count. Show why one fewer fails, why your count is sufficient, and what goes in the last envelope.

Answer: 17

Look for:

• Labels the quotient as a group count and the remainder as leftover objects before deciding.
• Uses the task's physical constraints to choose another container, a complete-group count or an equal fractional share.
• Independent rubric: one criterion each for 17 envelopes, 16 × 9 < 146 ≤ 17 × 9, and two cards in the final envelope; all three demonstrate the packing decision.

Teacher response: Sixteen envelopes hold only 144 cards. Seventeen hold up to 153, so a seventeenth envelope carries the last two cards. A complete explanation labels 17 envelopes and two cards separately.

transfer. A 137-metre ribbon can be cut without loss and must be divided into eight equal lengths. What length does each person receive? Explain why neither 18 metres nor 17 metres with one metre discarded meets this job.

Answer: Each receives 17 1/8 metres. Eight 17-metre lengths use 136 metres; the remaining metre is cut into eight equal eighth-metre pieces. Thus 8 × 17 1/8 = 137. Eighteen metres each would require 144 metres, and discarding the metre would fail to share the entire ribbon.

teach-1. 5 minutes: Have everyone complete the diagnostic without help. Ask two learners to label the units in 53 = 6 × 8 + 5. Use the routing decision to provide a brief grouping model or move directly to the main question.

teach-2. 10 minutes: Pairs draw the 137-card packing record using a labelled box for each full envelope rather than drawing every card. One partner proposes an envelope count; the other tests its capacity. Model the last envelope and compare the complete-set interpretation.

teach-3. 15 minutes: Spend five minutes on the support or core task according to the diagnostic, six minutes comparing the two routes and attempting the challenge, and four minutes recording a sentence that links each decision to its constraint. Require an equation and units, then switch checker and solver roles.

teach-4. 10 minutes: Allow six minutes for the independent check without a partner, then four minutes for the ribbon transfer and discussion. Assess three exit criteria: correct container count, an insufficient/sufficient capacity comparison, and an explanation of what happens to the leftover. Reteach the missing criterion before another independent packing problem.

misconception. Always add one whenever division leaves a remainder.

Teacher response: Ask what must happen to the leftover. Another container is needed when every indivisible card must travel. A complete-set count keeps the spare outside, while a divisible length can be shared in fractional units. Name the object being counted before choosing an answer.

support. Use 29 counters and four drawn envelopes. Fill three, count the five outside and ask whether those five are allowed to remain behind. Then repeat with the complete-kit condition before using larger numbers.

extend. Find two totals that need 18 eight-card envelopes but leave different numbers of spaces unused. For example, 137 cards leave seven spaces and 144 leave none; show that both still use 18 envelopes.

support-route. If the learner does not obtain five or calls it five groups, draw eight boxes of six, count the five outside, and label groups and counters separately. Recheck with 20 = 6 × 3 + 2 before the support task.

ready-route. If the learner records eight groups and five leftover counters with matching units, begin the worked packing decision and ask which constraint determines the final answer.