Target: calculate a three-set union using pairwise intersections that include the triple intersection. Prepare: C, three-set region map.
- Launch · 2 min. Ask how often a person in all three is counted by
12+11+9. - 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. - 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.
- Practice route · 7 min. Choose Day 3 route; compute union both by formula and region totals, marking whether pair counts include the centre.
- 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. - Exit · 3 min. “Why add 2 at the end?” Expected: it restores people counted zero times after the pair subtractions.