Find when two delivery cycles meet again

diagnostic. A cart returns every four minutes after a shared start. Enter its third positive return time in elapsed minutes, and list its first three returns.

Answer: 12

Teacher response: The returns after the start are 4, 8 and 12 minutes. Zero is the starting event, not the first positive return.

model. Two supply carts are together at a classroom doorway at 8:00 a.m. The red cart returns every six minutes and the blue cart every eight minutes. After 8:00, when are they next at the doorway together? Give both elapsed minutes and the clock time.

1. Mark the shared start as zero elapsed minutes and label it 8:00 a.m. Circle the word next: the meeting at zero is already known and is excluded. A return time must be a whole number of each cart's complete cycles.
2. List red returns in increasing order: 6, 12, 18, 24. List blue returns up to the same point: 8, 16, 24. Continue every list without skipping a cycle so an earlier shared return cannot be hidden.
3. Circle 24 in both lists. Confirm that 24 ÷ 6 = 4 and 24 ÷ 8 = 3, so both carts return then. None of 6, 12 or 18 occurs in the blue list, so no earlier positive red return is shared. This establishes the first meeting.
4. Convert the elapsed result using the common start: 24 minutes after 8:00 is 8:24. Keep both labels in the answer. Continue the cycles to 48 minutes if checking the next meeting, and explain why a different starting time would require rebuilding that cart's return list.

Worked example: They next meet after 24 minutes, at 8:24 a.m. The red cart has completed four six-minute cycles and the blue cart three eight-minute cycles: 4 × 6 = 3 × 8 = 24.

p1. Support: two carts are together at 9:00 a.m. and return every four and six minutes. Enter the elapsed minutes until their next meeting and list both sets of returns up to that time.

Answer: 12

Teacher response: The four-minute returns are 4, 8, 12; the six-minute returns are 6, 12. The first shared positive value is 12, at 9:12 a.m.

p2. Core: two carts start together at 10:00 a.m. and return every six and nine minutes. Enter the elapsed minutes to the next meeting, record the clock time, and show that no earlier return is shared.

Answer: 18

Teacher response: The lists 6, 12, 18 and 9, 18 first share 18. The next meeting is at 10:18 a.m.; two nine-minute cycles and three six-minute cycles have passed.

p3. Challenge: two carts start together at 11:10 a.m. and return every four and six minutes. How many meetings occur after the start and up to 11:46 a.m., including a meeting at the endpoint? Enter the count and list the meeting clock times.

Answer: 3

Teacher response: The window lasts 36 minutes. Meetings occur every 12 minutes after the shared start: at 11:22, 11:34 and 11:46. The start is excluded and the endpoint is included.

e1. Independent check: two carts are together at 1:00 p.m. and return every eight and five minutes. Enter the elapsed minutes until their next meeting. List both sets of returns through that meeting, give its clock time, and justify why it is the first.

Answer: 40

Look for:

• Generates complete multiples of both cycle lengths and excludes the shared starting event when asked for the next meeting.
• Proves earliest by checking all earlier returns of one cart against the other list.
• Independent rubric: one criterion each for 40 minutes, complete lists with no smaller shared positive entry, and 1:40 p.m.; all three demonstrate coordination from a shared start.
• Checks a changed starting time before reusing a common-multiple result.

Teacher response: The lists are 8, 16, 24, 32, 40 and 5, 10, 15, 20, 25, 30, 35, 40. They first share 40, giving 1:40 p.m. Five eight-minute cycles and eight five-minute cycles agree.

transfer. Keep the red cart's six-minute cycle starting at 8:00, but let the blue cart begin its eight-minute cycle at 8:03. Are both at the doorway at 8:24? Explain using each actual starting time.

Answer: No. Red returns at 8:06, 8:12, 8:18 and 8:24. Blue is at the doorway at its start of 8:03 and returns at 8:11, 8:19 and 8:27. The earlier common-multiple table relied on a shared start; shifting blue's start changes its clock times.

teach-1. 5 minutes: Use the diagnostic to check equal-step counting and whether zero is being counted as a return. Route support learners to a jump line and ask ready learners to distinguish elapsed minutes from a clock reading.

teach-2. 10 minutes: Partners build the red and blue return lists in different colours. Stop when a common value appears, verify it by division, and cross-check every earlier red return against the blue list. Together label the result on the clock.

teach-3. 15 minutes: Spend five minutes on the support or core pair of cycles, five minutes checking earliest-return evidence with a partner, and five minutes on the inclusive counting challenge. Require a shared starting event and units in each record; distinguish first meeting from number of meetings.

teach-4. 10 minutes: Allow six minutes for the independent return lists and clock time, then four for the changed-start transfer. Assess three exit criteria: first common positive multiple, complete earlier-return evidence and correct clock label. Revisit equal jumps for skipped returns or the start marker for a clock-time error.

misconception. Multiply the two cycle lengths, so the next meeting must be after 6 × 8 = 48 minutes.

Teacher response: Forty-eight is a shared return, but test whether a smaller shared return appears. Listing both sets finds 24 first. Ask for the earliest shared positive time and use the shorter lists to justify why it is earliest.

support. Use two colours to mark four-minute and six-minute jumps from the same zero on a minute line. Say the event at zero aloud before counting positive returns; circle the first later mark receiving both colours.

extend. For the main six- and eight-minute cycles, list the next two meetings after 8:24: 8:48 and 9:12. Show that adding 24 minutes preserves a whole number of both cycles, and state both elapsed and clock values.

support-route. If the learner gives 8 or includes zero as the first return, draw equal four-minute jumps and label the starting event separately. Recheck returns at 5, 10 and 15 for a five-minute cycle before the support task.

ready-route. If the learner gives 12 and lists 4, 8, 12, begin the two-cycle table and ask which elapsed values occur in both lists.