# Ten 25-minute teacher scripts · counting techniques opener

Every lesson has **2 + 4 + 5 + 7 + 4 + 3 = 25 minutes**. Use [A–F practice files](CASE-CARDS.md), [learner page](LEARNER.md), [three same-target routes](DAILY-CHOICES.md) and an [original A4 aid](print/TEXT-ALTERNATIVES.md). Every count needs an interpretation and a reason the chosen operation avoids omissions or duplicates. Speak notation as words when useful: `|A∪B|` = “number in A or B or both”; `|A∩B|` = “number in both.” Read, draw, move raised region cards, type or speak without changing the target. A 7-minute response window on check days can be extended locally; speed is not the mathematics being assessed. Fresh G/H/I are given only on their check days; the separate key is used afterward.

### Day 1 · Count each member once

**Target:** derive and apply two-set inclusion–exclusion for a union when both set counts and overlap are known. **Prepare:** A, two-set map.

1. **Launch · 2 min.** Ask whether 17 sketching plus 13 coding equals the number of distinct people in A.
2. **Model · 4 min.** Put 5 “both” tokens into the overlapping region first. Then 12 sketch-only and 8 coding-only. The 5 both-tokens appeared in each set total, so `17+13−5=25` distinct participants.
3. **Guided reading · 5 min.** Use A's universe 30. Add 5 neither; four disjoint regions sum 30. Explain why the neither group is outside the union, not inside a circle.
4. **Practice route · 7 min.** Choose Day 1 route; give formula, region reconstruction, answer with people as units, and one sentence about double count.
5. **Audit · 4 min.** Show claim `17+13=30 distinct participants`. Learner locates the 5 repeated names and repairs to 25.
6. **Exit · 3 min.** “What is the union of A's two groups?” Expected: 25 people; 17+13−5 because both were counted twice.

### Day 2 · Solve for an unseen overlap

**Target:** rearrange the two-set rule to recover an intersection and then a neither count. **Prepare:** B, two-set map.

1. **Launch · 2 min.** State B's 40 total and 32 in at least one. Ask how many lie outside both before any formula.
2. **Model · 4 min.** `32=22+18−x`, so `x=8` in both. Neither `40−32=8`. Regions drawing-only 14 and music-only 10; `14+8+10+8=40`.
3. **Guided reading · 5 min.** Ask why `22+18−32` is nonnegative and no larger than 18. Use 8 to verify each original set count.
4. **Practice route · 7 min.** Choose Day 2 route; solve x and neither, then use a four-region audit to explain reasonableness.
5. **Audit · 4 min.** Repair “8 neither means 8 both.” They happen to match numerically here but represent different regions; a changed universe would separate them.
6. **Exit · 3 min.** “Does B's 8 both include either single-group region?” Expected: no; those 8 selected both activities.

### Day 3 · Three sets: add the centre back

**Target:** calculate a three-set union using pairwise intersections that include the triple intersection. **Prepare:** C, three-set region map.

1. **Launch · 2 min.** Ask how often a person in all three is counted by `12+11+9`.
2. **Model · 4 min.** Show the seven regions. Sum the three sets 32, subtract pair overlaps 4+3+5=12, then add all-three 2 once: `32−12+2=22`. A triple member was counted 3 times, subtracted 3 times, then restored once.
3. **Guided reading · 5 min.** Reconstruct A&R-only=4−2=2, A&C-only=3−2=1, R&C-only=5−2=3; singles 7,4,3. These seven disjoint counts sum 22; universe 28 leaves 6 in none.
4. **Practice route · 7 min.** Choose Day 3 route; compute union both by formula and region totals, marking whether pair counts include the centre.
5. **Audit · 4 min.** Repair `32−12=20`; the two all-three people would then be missing. Repair counting all pair overlaps as “only” by subtracting the centre first.
6. **Exit · 3 min.** “Why add 2 at the end?” Expected: it restores people counted zero times after the pair subtractions.

### Day 4 · One route OR another

**Target:** apply the addition principle only when alternatives are disjoint, and identify when overlap would break a plain sum. **Prepare:** D, sum-or-product chooser.

1. **Launch · 2 min.** Ask whether one D participant books a studio session and an outdoor session. In this case they book exactly one.
2. **Model · 4 min.** List S1/S2/S3 and O1/O2/O3/O4. Seven distinct outcomes exist: `3+4=7`. A product 12 would describe a pair of sessions, not the stated one-session booking.
3. **Guided reading · 5 min.** Imagine one session appears on both lists. Name it once in the union; add the two list counts then subtract the shared option. Thus the addition principle's plain sum needs exclusive alternatives.
4. **Practice route · 7 min.** Choose Day 4 route; count D, write the outcome set, and explain why 3×4 is wrong in this model.
5. **Audit · 4 min.** Repair “OR always means add” using A: overlapping interests require subtraction.
6. **Exit · 3 min.** “What would 3×4 count in D?” Expected: a pair with one studio and one outdoor session, a different decision.

### Day 5 · Fresh public Check A: File G

**Target:** independently transfer two-set union/neither reasoning and disjoint addition to [new G](STUDENT-CHECKS.md#day-5-check-a--file-g). **Prepare:** blank two-set map and sum-or-product chooser; neutral access log.

1. **Launch · 2 min.** Explain that G is a new fictional case; its first response will inform the next lesson.
2. **Source access · 4 min.** Give only File G. Read numbers or notation neutrally if requested, without naming a calculation.
3. **Independent plan · 5 min.** Learner chooses regions/operation and marks what “neither” and “one booking” mean.
4. **Independent response · 7 min.** Choose Day 5 route; retain the initial calculation and explanation before content prompts.
5. **Self-audit · 4 min.** Learner checks that the both-region is not doubled and a one-session booking is not a pair.
6. **Submit · 3 min.** Collect response/access log; use public key for feedback, not a QCAA unit result.

### Day 6 · One choice AND another

**Target:** use the multiplication principle for two independent stages and explain why each first choice pairs with each second choice. **Prepare:** E, branch tree aid.

1. **Launch · 2 min.** Ask whether one finished E label needs a shape, a caption, or both.
2. **Model · 4 min.** Draw three shape branches; each has four caption leaves. Count leaves as 4+4+4=12, then compress to `3×4=12`.
3. **Guided reading · 5 min.** Check a particular shape pairs with all four captions. Ask what changes if a shape cannot take one caption: the uniform product would overcount and branch counts must be added instead.
4. **Practice route · 7 min.** Choose Day 6 route; show a tree/table or grouping, product and justification, with “finished labels” as units.
5. **Audit · 4 min.** Repair `3+4=7`: it counts single choices from either list, not a two-part label.
6. **Exit · 3 min.** “Why is 3×4 valid for E?” Expected: every one of three shapes can pair with every one of four captions.

### Day 7 · Branch first; count within each branch

**Target:** combine addition across exclusive branches with multiplication inside each branch. **Prepare:** F, branch tree and method chooser.

1. **Launch · 2 min.** Ask whether F chooses visual and audio together. It chooses exactly one route.
2. **Model · 4 min.** Visual route `2 themes×3 layouts=6`; audio route `2 narrations×2 paces=4`; exclusive routes sum to `6+4=10`.
3. **Guided reading · 5 min.** Enumerate visual leaves V11–V23 and audio leaves A11–A22. All ten are unique. Explain that `2×3×2×2=24` would require a label with all four attributes, which F does not.
4. **Practice route · 7 min.** Choose Day 7 route; draw or say branch counts, total and two decision words: AND inside, OR across.
5. **Audit · 4 min.** Repair `6×4=24`; it counts a visual-audio pair, not one selected route.
6. **Exit · 3 min.** “What is F's number of complete choices?” Expected: 10, assuming route categories stay exclusive.

### Day 8 · Check all seven regions

**Target:** verify a three-set calculation by reconstructing exclusive regions and rejecting an impossible set claim. **Prepare:** C, region audit mat.

1. **Launch · 2 min.** Ask whether a pairwise count of 4 in C means exactly 4 in A&R only. It includes the two all-three people.
2. **Model · 4 min.** Centre 2; pair-only 2,1,3; single-only 7,4,3. Sum seven regions `2+2+1+3+7+4+3=22`. Check A=7+2+1+2=12, R=4+2+3+2=11, C=3+1+3+2=9.
3. **Guided reading · 5 min.** Propose a claim “A&R pair=1 while all-three=2.” It is impossible because the triple is a subset of the pair. Discuss that nonnegative region counts and subset bounds are reasonableness tests.
4. **Practice route · 7 min.** Choose Day 8 route; reconstruct and audit C's seven cells and reject one impossible alternative.
5. **Audit · 4 min.** Repair any negative region by revisiting whether a pair count already includes all-three.
6. **Exit · 3 min.** “What is the minimum possible size of each pair in C?” Expected: at least 2, the all-three count; actual pairs are 4,3,5.

### Day 9 · Pick a method and defend it

**Target:** classify whether a new counting story needs inclusion–exclusion, disjoint addition, multiplication, or a branch combination, and justify the classification. **Prepare:** A/D/E/F, method chooser.

1. **Launch · 2 min.** Put `both`, `one of`, `one from each` and `branch then pair` on cards. Ask which phrase hints at duplicate counting.
2. **Model · 4 min.** A overlaps → inclusion–exclusion 25. D one disjoint session → addition 7. E two stages → multiplication 12. F exclusive routes with stages inside → `2×3+2×2=10`.
3. **Guided reading · 5 min.** Ask what assumption each calculation requires and what a wrong operation would count instead. A needs pair overlap 5; E needs every shape-caption pairing.
4. **Practice route · 7 min.** Choose Day 9 route; sort four cards by method, supply one answer and one counterexample to an invalid operation.
5. **Audit · 4 min.** Challenge “just look for OR/AND.” Learner states the real test: mutually exclusive outcomes, overlapping sets, or stages that jointly form an outcome.
6. **Exit · 3 min.** “Can the same surface word OR always justify plain addition?” Expected: no; A's overlapping OR needs inclusion–exclusion.

### Day 10 · Fresh public Check B: Files H/I

**Target:** independently transfer three-set union and mixed branch counting to [new H/I](STUDENT-CHECKS.md#day-10-check-b--files-h-and-i). **Prepare:** blank three-set map and branch tree; neutral support log.

1. **Launch · 2 min.** Explain that H/I are new fictional information and the response is public formative evidence.
2. **Source access · 4 min.** Give H/I only, with neutral reading or enlarged text as needed.
3. **Independent plan · 5 min.** Learner marks the triple, pair-only regions, exclusive branches and complete outcome units.
4. **Independent response · 7 min.** Choose Day 10 route; retain first formula, diagram and explanation before content help.
5. **Self-audit · 4 min.** Learner checks nonnegative regions, total universe and branch exclusivity.
6. **Submit · 3 min.** Collect work/support log; use separate key for next instruction, not a formal unit result.

SubjectNest original teacher scripts © NeuroForgeIO Pty Ltd 2026, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/).