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Year 11 / Specialist Mathematics / Term 1 / Weeks 01 02

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Ten 25-minute teacher scripts · counting techniques…Year 11 Specialist Maths · T1 W1–2 · Lesson sequence

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Teacher copy · prompts and answer keys

Teacher copy: This page may include teaching prompts or answer keys. Answer keys in this public library can be viewed by anyone. Give learners a clean prompt, use checks as formative evidence, and change a case locally when prior access matters.

Every lesson has 2 + 4 + 5 + 7 + 4 + 3 = 25 minutes. Use A–F practice files, learner page, three same-target routes and an original A4 aid. 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. 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. 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.

Source: https://neuroforge.io/subjectnest/library/resources/year-11/specialist-mathematics/term-1/weeks-01-02/LESSONS · Original material © NeuroForgeIO Pty Ltd 2026 · CC BY 4.0: https://creativecommons.org/licenses/by/4.0/ · Credit the source and note changes. External sources keep their own terms.