Groups, quotients and decisions

pre1. For 132 × 24, a draft correctly finds 132 × 4 = 528 but writes 264 as the row for 132 × 20. Enter the corrected value of that second row and explain what the 2 in 24 represents.

Answer: 2640

Look for:

• Names twenty groups and connects 264 tens with 2,640 ones. Record this explanation separately from the numeric response.

Teacher response: The 2 in 24 means two tens, or twenty groups. Since 132 × 2 = 264, twenty groups contain 2,640, not 264.

pre2. Each of 23 packs contains 148 labels. Enter the total number of labels. Write the partial products for twenty packs and three packs and explain why they are added.

Answer: 3404

Look for:

• Labels 2,960 as twenty packs and 444 as three packs, combines the non-overlapping groups, and names labels as the final unit. Record the grouping explanation separately from the total.

Teacher response: Twenty packs contain 2,960 labels and three contain 444. They account for all 23 equal packs, so 2,960 + 444 = 3,404 labels.

pre3. There are 83 indivisible cards. Each issued kit must contain exactly six cards; incomplete kits are not issued and spare cards remain in storage. Enter the number of complete kits. Write a quotient-and-remainder equation and state what happens to the spare cards.

Answer: 13

Look for:

• Produces or verifies 83 = 6 × 13 + 5 with 5 < 6, labels 13 as complete kits and five as stored cards, and does not count an incomplete kit. Note division errors separately from a wrong decision.

Teacher response: 83 = 6 × 13 + 5. Thirteen complete kits use 78 cards and leave five in storage. Five cards cannot make another six-card kit.

pre4. All 94 indivisible cards must travel in envelopes holding at most seven cards each. A partly filled final envelope is allowed. Enter the minimum envelope count, show why one fewer cannot carry every card and why your count can.

Answer: 14

Look for:

• Verifies 94 = 7 × 13 + 3, compares 91 < 94 ≤ 98, and links the extra envelope to the requirement that all cards travel. Record the capacity explanation separately from the count.

Teacher response: Thirteen envelopes hold 91 cards and leave three. A fourteenth envelope carries those three; its capacity brings the total to 98 spaces.

pre5. Two carts are together at a doorway at 10:00 a.m. and return every four and eight minutes from that same start. Enter the elapsed minutes until their next meeting after 10:00. List enough returns to show it is the first later meeting.

Answer: 8

Look for:

• Uses the shared zero, excludes the starting meeting, lists 4, 8 and 8, and explains that four minutes is not a return for both carts. Record the earliest-return evidence separately from the number.

Teacher response: The four-minute cart returns at 4 and 8 elapsed minutes; the eight-minute cart first returns at 8. Their next meeting is after eight minutes, at 10:08 a.m.

pre6. Two carts are together at 2:00 p.m. and return every three and five minutes from that same start. How many meetings occur after 2:00 and up to 2:45, including the endpoint? Enter the count and list their meeting times. Show each cart's returns through the first later meeting.

Answer: 3

Look for:

• Supports the 15-minute shared cycle with both return lists, identifies meetings at 15, 30 and 45 elapsed minutes, excludes zero and includes the endpoint. Record the event-count explanation separately from the count.

Teacher response: The first shared positive return is 15 minutes, and later shared returns in the 45-minute window are 30 and 45. The meetings are at 2:15, 2:30 and 2:45; the start is excluded.

post1. For 214 × 32, a draft correctly finds 214 × 2 = 428 but writes 642 as the row for 214 × 30. Enter the corrected value of that second row and explain what the 3 in 32 represents.

Answer: 6420

Look for:

• Names thirty groups and connects 642 tens with 6,420 ones. Record the tens explanation separately from the numeric response.

Teacher response: The 3 in 32 represents three tens, or thirty groups. 214 × 3 = 642, so 214 × 30 = 6,420.

post2. Each of 34 packs contains 126 labels. Enter the total number of labels. Write the partial products for thirty packs and four packs and explain why they are added.

Answer: 4284

Look for:

• Labels 3,780 as thirty packs and 504 as four packs, combines the non-overlapping groups and names labels as the final unit. Record the grouping explanation separately from the total.

Teacher response: Thirty packs contain 3,780 labels and four contain 504. Together they account for 34 packs, giving 3,780 + 504 = 4,284 labels.

post3. There are 97 indivisible cards. Each issued kit must contain exactly eight cards; incomplete kits are not issued and spare cards remain in storage. Enter the number of complete kits. Write a quotient-and-remainder equation and state what happens to the spare cards.

Answer: 12

Look for:

• Produces or verifies 97 = 8 × 12 + 1 with 1 < 8, labels 12 as complete kits and one as a stored card, and does not count an incomplete kit. Note division errors separately from a wrong decision.

Teacher response: 97 = 8 × 12 + 1. Twelve complete kits use 96 cards and leave one in storage. That card does not make another complete eight-card kit.

post4. All 119 indivisible cards must travel in envelopes holding at most eight cards each. A partly filled final envelope is allowed. Enter the minimum envelope count, show why one fewer cannot carry every card and why your count can.

Answer: 15

Look for:

• Verifies 119 = 8 × 14 + 7, compares 112 < 119 ≤ 120 and links the extra envelope to carrying every card. Record the capacity explanation separately from the count.

Teacher response: Fourteen envelopes hold 112 cards and leave seven. A fifteenth carries those seven; fifteen envelopes provide 120 spaces.

post5. Two carts are together at a doorway at 11:00 a.m. and return every three and nine minutes from that same start. Enter the elapsed minutes until their next meeting after 11:00. List enough returns to show it is the first later meeting.

Answer: 9

Look for:

• Uses the shared zero, excludes the starting meeting, lists 3, 6, 9 and 9, and explains that neither three nor six minutes is a return for both carts. Record earliest-return evidence separately from the number.

Teacher response: The three-minute cart returns at 3, 6 and 9 elapsed minutes; the nine-minute cart first returns at 9. Their next meeting is after nine minutes, at 11:09 a.m.

post6. Two carts are together at 9:00 a.m. and return every three and eight minutes from that same start. How many meetings occur after 9:00 and up to 9:48, including the endpoint? Enter the count and list their meeting times. Show each cart's returns through the first later meeting.

Answer: 2

Look for:

• Supports the 24-minute shared cycle with both return lists, identifies meetings at 24 and 48 elapsed minutes, excludes zero and includes the endpoint. Record the event-count explanation separately from the count.

Teacher response: The first shared positive return is 24 minutes. The shared returns in the 48-minute window are 24 and 48, so the meetings are at 9:24 and 9:48. The start is excluded.

transfer. An office opens 24 packs containing 36 cards each. All cards must travel in envelopes holding at most seven indivisible cards; a partly filled last envelope is allowed. Two carts are together at the office at 9:10 a.m. and return every four and nine minutes from that common start. The office will hand over the batch at their first meeting after the start. Write a decision note giving the total cards, minimum envelopes, cards in the last envelope, elapsed wait and handover clock time. Show a group-value calculation, an insufficient/sufficient capacity check and return-list evidence.

Answer: There are 864 cards, requiring 124 envelopes with three cards in the last envelope. The carts next meet after 36 minutes, at 9:46 a.m.

Teacher response: 24 × 36 = 720 + 144 = 864. Since 864 = 7 × 123 + 3, 123 envelopes hold only 861 cards, while 124 hold up to 868; the final three cards need one more envelope. Four-minute returns through 36 are 4, 8, 12, 16, 20, 24, 28, 32, 36; nine-minute returns are 9, 18, 27, 36. They first share 36, and 36 minutes after 9:10 is 9:46. For this transfer, record numeric outcomes and the three requested kinds of explanation separately, and note any missing evidence.

lesson-1. Find the place-value error in a product · 40 minutes. Before: Prepare place-value columns and the existing 246 × 32 worked audit. Administer pre1 and pre2 in the first five minutes. Use N to identify computation errors and E to identify a missing explanation of tens or equal groups; do not infer division readiness from either result. Focus: Use ten minutes to model the existing worked product and name the groups counted by each partial product. Use fifteen minutes for its support, core and challenge audit tasks, routing by the pre-assessment evidence and switching solver/checker roles. Reserve ten minutes for post1 and post2 plus an explanation of the tens row. The progression begins with the value of a group and a multiplier before using group counts in a division decision.

Teacher response: Evidence: Keep each learner's pre1/post1 tens-row explanation and pre2/post2 expanded product alongside the separate N/E records. Look for a multiplier such as 30 being named as thirty groups, correctly aligned partial products, and a total checked against the context. A correct total with an unexplained row remains an explanation gap. If secure: Move to the remainder investigation after its separate division entry check. Ask the learner to use multiplication to test a proposed group count, while keeping the requested quotient and remainder units explicit. If not secure: Use the group-value routing model and recheck before assigning another full audit. Keep the missing issue specific: tens value, within-row arithmetic, row alignment or final addition. Begin the next lesson with small equal groups if division is also insecure.

lesson-2. Choose what a remainder means for the job · 40 minutes. Before: Administer pre3 and pre4 in the opening five minutes and inspect the quotient-and-remainder equations, not only their final decisions. Confirm 24 ÷ 6 using equal groups when needed. Distinguish a basic-division gap from a learner who divides correctly but chooses the wrong response to the leftover. Focus: Use ten minutes for the existing 137-card worked decision and its comparison with complete sets. In fifteen minutes, use the investigation's differentiated tasks to require a division record, a named leftover unit and an explicit allowed action. Use the final ten minutes for post3 and post4 and a brief comparison of why complete kits and transport containers can require different decisions. Connect multiplication to checking capacity; keep division interpretation as a separate learning demand.

Teacher response: Evidence: Retain pre3/post3 records that identify full kits and stored spare cards, and pre4/post4 records comparing insufficient with sufficient container capacity. Record N and E separately. Look for a remainder smaller than the group size and a physical reason for keeping a spare outside or opening one more container. If secure: Move to the cycle investigation after its own multiples entry pair. Connect an exact return to a complete number of cycles, and ask why a partly completed cycle is not a return event. If not secure: If the quotient is wrong, return to equal-group division and verification before changing the decision. If the quotient is correct but the decision is wrong, use the remainder-decision routing model and recheck. Revisit all cards must travel versus only complete kits may be issued, then collect a fresh explanation.

lesson-3. Find when two delivery cycles meet again · 40 minutes. Before: Prepare a minute line with a clearly labelled zero and two colours. Administer pre5 and pre6 in the first five minutes. In every item both carts are together at the stated starting time. Check whether the learner generates every return and excludes zero when asked for a later meeting. Focus: Use ten minutes for the existing six- and eight-minute worked cycles. Spend fifteen minutes on its differentiated return-list tasks, verifying that a candidate time contains whole cycles of both lengths and that no earlier positive return is shared. Use the final ten minutes for post5 and post6 and the original changed-start discussion. Connect groups to multiples, quotients to completed cycles, and the decision to the earliest shared event after an explicit common start.

Teacher response: Evidence: Keep both coloured return lists, the first shared positive value and any counted meetings. Record N separately from E, which requires complete earlier-return evidence and correct treatment of starting and endpoint events. A product of the cycle lengths is only a candidate; the lists must establish the first shared return. If secure: Use the unit's combined transfer to inspect whether the learner carries units and constraints across the three settings. Review the work and explanation before choosing a further application; avoid treating a score total as proof of general mastery. If not secure: Use the cycle-coincidence routing line and recheck. Address skipped multiples, an included starting event or an unexamined candidate separately. If the start times differ, rebuild each actual event list before seeking a meeting.

route-1. group-value: Use place-value columns to label each multiplier part as a number of equal groups. For a wrong tens row, expand the multiplier before using the compact algorithm. If the row values are secure but the total is wrong, address the addition. If N is correct and E is missing, ask the learner to annotate the groups rather than repeating an unexplained computation. Recheck: Each of 26 packs holds 153 labels. Enter the total label count, show the rows for twenty packs and six packs, and explain their units.

Answer: 3978

Teacher response: Twenty packs contain 3,060 labels and six contain 918; 3,060 + 918 = 3,978 labels. Record N for 3,978 and E for the two correctly named groups and their combined label total.

route-2. remainder-decision: First inspect the division record. If it is wrong, model a smaller total with equal groups, verify quotient × group size + remainder, and check that the remainder is smaller than a group. Once basic division is available, contrast issuing complete kits with transporting all cards. If N is correct but E is missing, request both an insufficient-capacity and a sufficient-capacity statement. Recheck: All 78 indivisible cards must travel in envelopes holding at most five each; a partly filled final envelope is allowed. Enter the minimum envelope count, write the division record and compare its capacity with one fewer envelope.

Answer: 16

Teacher response: 78 = 5 × 15 + 3. Fifteen envelopes hold only 75 cards, while sixteen have 80 spaces and carry the final three cards. Record N for 16 and E for the division record, the two capacity checks and the all-cards constraint.

route-3. cycle-coincidence: Mark the shared starting event as zero, then make every positive return of each cart in a different colour. Check the earliest candidate against all earlier entries. If the starting event was counted, reread after the start; if an endpoint was missed, mark the inclusive endpoint explicitly. For a correct N with missing E, require the earlier-return evidence instead of accepting a product alone. Recheck: Two carts are together at 9:10 a.m. and return every two and six minutes from that same start. Enter the elapsed minutes until their next meeting after the start, write its clock time and justify why it is first.

Answer: 6

Teacher response: The two-minute returns are 2, 4 and 6; the six-minute cart first returns at 6. They meet next at 9:16 a.m. Record N for six minutes and E for both complete earlier lists, exclusion of zero and the matching clock time.

ready. Choose the combined transfer when the retained work shows correctly valued groups, a checked division record with a decision matching the physical constraint, and a first shared positive return from a common start. Review the numeric and explanation records together with the learner's work. These observations guide the next task; no total-score threshold is used to label general mastery.

support. Select the routing entry for each missing explanation or arithmetic step, then record N and E separately for its recheck. A correct number with missing reasoning receives a focused explanation prompt; an incorrect division record receives basic-division support before a remainder decision. Use the recheck evidence to choose the next example rather than repeating all three lessons.

extension. Ask the learner to revise one condition in the combined transfer: issue only complete seven-card kits, or shift one cart's starting time. Require a new decision record that identifies which calculations remain applicable and which event list must change. Use the revision as further evidence of transfer, without treating one successful response as a validated mastery measure.