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Fraction Classroom · 2.6.0 · Student pages

Choose what a remainder means for the job

Use the same division record to decide whether to open another container, keep a leftover or divide a measurable quantity into fractional shares.

Anonymous group code: __________   Date: __________

Each row uses 137 ÷ 8. The ribbon row means eight equal shares of 137 metres; the card rows mean groups of up to, or exactly, eight cards. The requested quantity changes.JobFull groupsLeftoverDecisionTransport all17 envelopes1 card18 envelopesComplete sets17 sets1 spare card17 setsShare ribbon17 m each1 m to share17 1/8 m each
Each row uses 137 ÷ 8. The ribbon row means eight equal shares of 137 metres; the card rows mean groups of up to, or exactly, eight cards. The requested quantity changes.

Start with a diagnostic

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.

Build the idea

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.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

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.

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.

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.

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.

Transfer the idea

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.