A three-lesson teacher plan connecting place-value multiplication, division decisions and repeating cycles through equal groups. Reuse three existing 40-minute investigations with parallel entry and exit evidence.
Get assessments and teacher materialsGoal: Help learners name what a group represents, verify the relevant arithmetic and explain how units, physical constraints and a shared starting event determine a decision.
Materials: The existing investigations Find the place-value error in a product, Choose what a remainder means for the job, and Find when two delivery cycles meet again; paper, pencil, a place-value grid, two coloured pencils, a minute line and up to 30 counters per pair. Use drawn or labelled groups for larger quantities. Keep an individual evidence record with a separate N and E entry for each assessment item.
Use the relevant two pre-assessment items in the opening five minutes of each lesson: pre1–pre2 before the product audit, pre3–pre4 before the remainder decisions, and pre5–pre6 before the cycles. Use the matching post-assessment pair in that lesson's final ten minutes. For every item record N = 1 when the numeric response exactly matches the stated quantity, otherwise N = 0; record E = 1 when the learner's written or spoken reasoning supplies the specific evidence in lookFor, otherwise E = 0 and note the missing evidence. Record these two values separately and retain the work; a correct number alone does not establish the explanation. Use patterns of missing evidence to select support, rather than using a total-score cutoff to declare mastery. Before remainder work, independently check basic division: ask the learner to form 24 ÷ 6 = 4 and to verify a quotient-and-remainder record with multiplication and addition. Multiplication supports checking, but a secure product audit alone does not establish division understanding. If grouping or the division record is insecure, model equal groups before asking what a remainder means. Before cycles, check that the learner can generate complete positive multiples from the same zero and distinguish the shared starting event from the next return.
Use these six questions independently. Keep the numerical answer and the requested explanation as separate observations.
2640 · The 2 in 24 means two tens, or twenty groups. Since 132 × 2 = 264, twenty groups contain 2,640, not 264.
Reasoning evidence: Names twenty groups and connects 264 tens with 2,640 ones. Record this explanation separately from the numeric response.
3404 · Twenty packs contain 2,960 labels and three contain 444. They account for all 23 equal packs, so 2,960 + 444 = 3,404 labels.
Reasoning evidence: 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.
13 · 83 = 6 × 13 + 5. Thirteen complete kits use 78 cards and leave five in storage. Five cards cannot make another six-card kit.
Reasoning evidence: 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.
14 · Thirteen envelopes hold 91 cards and leave three. A fourteenth envelope carries those three; its capacity brings the total to 98 spaces.
Reasoning evidence: 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.
8 · 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.
Reasoning evidence: 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.
3 · 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.
Reasoning evidence: 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.
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.
Collect: 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 more support is needed: 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.
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.
Collect: 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 more support is needed: 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.
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.
Collect: 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 more support is needed: 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.
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.
Revisit the model and teaching sequence
Fresh check: 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.
3978 · 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.
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.
Revisit the model and teaching sequence
Fresh check: 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.
16 · 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.
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.
Revisit the model and teaching sequence
Fresh check: 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.
6 · 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.
The skills match the first set; the quantities change. Compare the explanation for each skill as well as the number.
6420 · The 3 in 32 represents three tens, or thirty groups. 214 × 3 = 642, so 214 × 30 = 6,420.
Reasoning evidence: Names thirty groups and connects 642 tens with 6,420 ones. Record the tens explanation separately from the numeric response.
4284 · Thirty packs contain 3,780 labels and four contain 504. Together they account for 34 packs, giving 3,780 + 504 = 4,284 labels.
Reasoning evidence: 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.
12 · 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.
Reasoning evidence: 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.
15 · Fourteen envelopes hold 112 cards and leave seven. A fifteenth carries those seven; fifteen envelopes provide 120 spaces.
Reasoning evidence: 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.
9 · 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.
Reasoning evidence: 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.
2 · 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.
Reasoning evidence: 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.
Ready to continue: 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.
More 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.
Extend: 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.
These original checks support teaching decisions. They are not a standardised assessment or a measure of classroom effectiveness.
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.
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. · 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.
Use an anonymous group code. For each question, record numerical correctness and the requested explanation in separate columns. Leave a short next step tied to the skill.
Download the blank group record (CSV)The file contains blank assessment rows. Keep any completed classroom record in your own storage.
Student pages contain both assessment sets. Teacher pages add answers, evidence criteria, lesson routing and fresh checks.
This plan connects the linked lessons. Each lesson records its curriculum scope and original models; the sequence adds parallel checks and teaching decisions.
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