4. Sample: Filled Lesson Plan (60 min) — Calculating Combinations (no probabilities)

Title: Calculating combinations (without probabilities)
Grade: 8 / Maths
Duration: 60 min

Learning objective:

• Primary: Calculate the number of possible ordered letter combinations when elements are distinct (e.g., compute 6 × 5 × 4 for three‑letter permutations without repetition).
• Secondary: Explain in one sentence why multiplicative counting applies.

Why:

• Curriculum: combinatorics unit; prerequisite for probability and counting arguments.
• Real-life: arrangements, passwords, simple combinatorial planning.

Diagnostic starting level (ZPD):

• Quick 3‑question diagnostic on mini-cards (prior: multiply 2-digit × 1-digit; counting by listing).
• If >80% correct on simple multiplication but unfamiliar with counting patterns -> ready for permutation structure.

Materials:

• Letter cards, OneNote page (“Letter Combinations”), whiteboard, group worksheets, exit tickets.

Timing & sequence:

1. Activation / diagnostic (10 min)
   • Activity: Letter-combination warm-up. Give groups 4 letter sets (E I T; A I H; etc.). Ask: “Form all 3-letter combinations; count them.” (Inspection triangle / Headlines)
   • Teacher collects one example per group on OneNote. Purpose: surface prior strategies and misconceptions.
   Cognitive load tactic: concrete manipulatives reduce abstraction; short timed task keeps focus.
2. Direct instruction + worked example (15 min)
   • Teacher demonstrates on board: two approaches — listing (low N) and rule-of-product (6 × 5 × 4). Write steps and explain why each factor decreases when repetition is not allowed (intrinsic load managed by example).
   • Show a worked example: 3 letters from 6 distinct letters (6×5×4). Include a contrasting worked example with repetition allowed (6×6×6) to illustrate rule boundary.
   ZPD scaffold: sentence starters — “I multiply because…” and “I cannot reuse letters when…”
3. Guided practice (pairs/small groups) (20 min)
   • Tasks: set of 4 problems of increasing complexity (with/without repetition; mixing types).
   • Teacher circulates, gives prompts: “Which choice in the first slot determines the next?” Use pair roles: explainer & checker (peer tutoring).
   • Checkpoints every 7 minutes: quick show of whiteboards / thumbs.
   Cognitive load tactic: fading support — begin with full scaffold, gradually remove as confidence grows.
4. Challenge / consolidation (5–7 min)
   • Whole-group puzzle: “Matti places 4 different fruit candies in a bag one by one. In how many sequences can they be placed?” (order matters).
   • Students answer on post‑it; teacher collects one exemplar solution.
5. Reflection & formative assessment (5 min)
   • Exit ticket: compute a short 2‑item task and write one sentence justification.
   • Criteria: correct numeric answer + correct reasoning phrase indicates mastery.
6. Closure & homework (3 min)
   • Review “key idea” (WHY + formula).
   • Homework: page ref. problems 352, 353; model solutions placed in OneNote. Tell students how homework will be checked next class.

Success criteria:

• Proficient: answers correct AND justification shows multiplicative reasoning.
• Developing: correct numeric answer but partial justification.
• Beginning: incorrect answer and no reasoning.

Backup plan:

• If many struggle: split class into two groups; reteach one worked example with extra scaffolds.
• If many finish early: extension using combinations where order doesn’t matter (introduce C(n,k) for motivated students).

Annexes:

• Full task texts and worked solutions attached to OneNote pages for teacher and student use.