A correct group answer does not establish that every member can explain it. This original study-group packet makes four actions visible: proposing a reason, questioning it, checking it against supplied facts and recording the correction. An individual response then shows what each participant can explain without a peer supplying the reasoning.
All names, stock records and dialogue are fictional. Four people and the suggested timings are convenient for this illustration, not universal rules about ideal groups. The teacher observes how explanations are checked; the kit arithmetic is a shared context rather than a separate curriculum claim.
The complete learner packet
Each completed kit requires exactly one blue card and two white tags. Only undamaged tags may be used. A tag cannot be divided or reused, and incomplete kits do not count. No extra stock arrives. For every task, find the maximum complete kits, explain both stock limits and test whether one additional kit is possible. Keep this rule and all task facts visible throughout.
| Task | Blue cards | Total white tags | Damaged tags |
|---|---|---|---|
| A | 14 | 23 | 3 |
| B | 9 | 25 | 1 |
| C | 12 | 27 | not supplied |
| D | 11 | 25 | 4 |
| I: final individual | 18 | 31 | 5 |
Before discussion, each person writes a brief attempt for the current round. They may consult the facts and rule, but should not copy another person’s explanation. Keep the attempt even if it is wrong or incomplete. After discussion, write a separate group explanation. Reveal the reference card only after the group’s attempt, then explain any correction rather than replacing the answer silently.
A completed plan with rotating responsibilities
Allow an illustrative three minutes to read the rule, four short rounds of about four minutes, and five minutes for the final individual response and teacher check. Extend or simplify when access or explanation needs more time. Read the task aloud or offer accessible text and a communication aid without hiding the rule or numerical facts.
| Round | Explainer | Questioner | Checker | Recorder |
|---|---|---|---|---|
| A | Ari | Beka | Chao | Deni |
| B | Beka | Chao | Deni | Ari |
| C | Chao | Deni | Ari | Beka |
| D | Deni | Ari | Beka | Chao |
- Explainer: “Cards allow this many; usable tags allow this many; the smaller limit controls.” Show the quantities, not only the final count.
- Questioner: “What happens to damaged tags? Can we build one more kit?” Test a specific claim instead of asking only whether everyone agrees.
- Checker: Independently recompute usable tags and required stock, then compare with the revealed reference card.
- Recorder: Write the original claim, decisive check and revised reason; explain aloud or in writing why the additional kit fails. Recording is reasoning, not copying.
Each participant takes every role across the four rounds, then completes the same final task individually. The teacher may interrupt dominance or clarify an unreadable instruction. Any explanation supplied by the teacher is recorded as support.
Worked case 1: a confident wrong solver dominates
Ari: “Fourteen kits. There are fourteen blue cards. Write that.” Beka: “Fourteen would need twenty-eight undamaged tags. How many do we have?” Ari: “Twenty-three.” Teacher: “Pause and give Chao the stock check.” Chao: “Three are damaged, so only twenty are usable. Twenty tags make ten pairs.” Deni: “I will keep the first claim and record ten kits, not erase the disagreement.”
The revised note is: “Initial claim: 14. Check: 23 − 3 = 20 usable tags; 14 kits would require 28. Corrected group result: 10 kits, 4 blue cards and 0 usable tags left. An eleventh kit requires 22 usable tags, more than 20.” Deni explains that last check; a copied number alone would not demonstrate the recorder’s reasoning.
Worked case 2: unanimous but unjustified
Round B gives 9 cards, 25 tags and 1 damaged tag. Beka: “Nine.” All: “Agreed.” Chao: “Agreement is not our check. Which stock runs out?” Deni: “Twenty-four usable tags could supply twelve kits, but only nine cards are available.” Ari: “Our note should say nine kits use nine cards and eighteen tags, leaving six usable tags.” Beka: “A tenth kit fails because no card remains.”
The number was correct before the discussion, but the reason was missing. Keep “9, no reason given” as the initial record and the stock comparison as peer-assisted evidence. Do not convert the whole exchange into four independent explanations.
Worked case 3: a missing fact needs escalation
Round C gives 12 cards and 27 tags but omits the damaged-tag count. Chao: “Most of us think twelve.” Deni: “If none are damaged, twelve works; if seven are damaged, only ten works.” Ari: “Both fit the supplied facts. We need the missing count.” Teacher: “Record the question; the packet intentionally does not supply an exact value.” Beka: “No unique result. Ask how many of the twenty-seven tags are damaged.”
The revised note contains two checks: with zero damaged, 27 usable tags and 12 cards allow 12 kits; with seven damaged, 20 usable tags allow 10. A vote cannot create the missing fact. The possible maximum-kit result ranges from 0 to 12 if the damaged count could be any whole number from 0 to 27.
Reference cards: reveal after attempts
| Task | Stock check | Maximum and remainder | One-more check |
|---|---|---|---|
| A | 23 − 3 = 20 usable; 20 ÷ 2 = 10 pairs; 14 cards | 10 kits; 4 cards; 0 usable tags | 11 require 22 usable tags, exceeding 20 |
| B | 25 − 1 = 24 usable; 24 ÷ 2 = 12 pairs; 9 cards | 9 kits; 0 cards; 6 usable tags | 10 require 10 cards, exceeding 9 |
| C | Damaged count is unknown | No unique answer; possible result 0–12 kits | Ask for damaged count; test conditional cases |
| D | 25 − 4 = 21 usable; 10 full pairs plus 1 tag; 11 cards | 10 kits; 1 card; 1 usable tag | 11 require 22 usable tags, exceeding 21 |
Round D gives the fourth participant a turn as explainer. It also makes an odd leftover visible: one usable tag remains, but one tag cannot form the required pair. The checker must confirm both feasibility and maximality: ten kits can be made, and eleven cannot.
Keep final individual evidence separate
Final task I uses 18 cards and 31 tags, 5 damaged. Everyone retains the rule and facts and writes without peer help. The checked result is 13 kits: 26 usable tags make 13 pairs, and 18 cards are sufficient. Five cards and no usable tags remain. Fourteen kits would require 28 usable tags.
| Participant | Individual response | Teacher interpretation and next check |
|---|---|---|
| Ari | 31 − 5 = 26 usable; 26 ÷ 2 = 13; 18 cards suffice | Both limits explained; ask to name remaining cards |
| Beka | 13 because 26 tags make 13 pairs | Tag limit explained; ask how card stock was checked |
| Chao | 15 because 31 tags make 15 pairs | Damaged tags not removed; return to the usable-tag rule |
| Deni | 13; 14 would need 28 usable tags but only 26 remain; 18 cards are enough | Feasibility and one-more test supplied |
Return Chao’s response with a question about which tags meet the rule, then collect a new explanation and label the prompt. The group’s earlier correct notes do not cancel this missing distinction. Conversely, one incomplete response does not justify a fixed ability label. No numerical group-work grade is assigned here.
Three fresh practice decisions
- A new task has 7 cards and 19 tags, 2 damaged. The recorder says “8 kits because 17 tags make 8 pairs.” What should the checker ask?
- For 15 cards and 26 tags, 4 damaged, the group writes “11 kits, everyone understands.” What evidence is still missing?
- A card says 10 blue cards and “at least 16 usable tags.” May the group report exactly 8 kits?
Teacher check: explained answers
- Ask how many cards eight kits require. There are only seven. The maximum is 7, using 14 of the 17 usable tags and leaving 3. The tag-only calculation ignores the tighter card limit.
- The count is right: 22 usable tags allow 11 kits and leave 4 cards. Collect each person’s independent explanation before claiming individual understanding. Preserve the distinction between group checking and individual work.
- No unique maximum follows. Sixteen usable tags allow 8 kits, eighteen allow 9, and twenty or more allow 10. Ask for the actual usable-tag count or report the bounded result of 8–10.
Frequently asked questions
Why keep the rule visible?
The target is checking explanations and sharing responsibility. This is not a closed-book memory task.
Why rotate the recorder?
Every participant should practise documenting and explaining a decisive check, as well as proposing the original solution.
Does agreement count as verification?
No. Verification applies the rule to the quantities and tests whether one extra kit is possible.
Must all groups have four members?
No. Four makes this role rotation explicit. Adapt participation and task demands to the actual group.
Can a group say there is not enough information?
Yes, when it identifies the missing fact and shows how different possible values change the answer.
When should the reference card appear?
After the attempt and peer check. Learners should explain why their record agrees or needs correction.
What does the final individual task establish?
It supplies immediate evidence of that person’s reasoning under the stated conditions, not proof of long-term retention or broad ability.
Sources and scope
These sources inform the teaching approach; the story, packet, dialogue and records are original illustrations. Cornell University Center for Teaching Innovation — Plan each stage. Carnegie Mellon University Eberly Center — Articulate expectations. No official examination grading or current Kenyan grade allocation is implied.
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