Canonical combined lesson MATH30B-L044 · week 9 day 4 · CURRICULUM_BASELINE

MATH30B-L044 - Double-angle identities; odds in context; probability problems

• Delivery id: MATH30B (combined classroom; not a fourth credential)
• Official courses: Mathematics 30-1 (MAT3791), 30-2 (MAT3792), 30-3 (MAT3793)
• Lesson ID: MATH30B-L044
• Calendar: Week 9 Day 4 of 18 weeks × 5 days (lesson 44 of 90)
• Object type: canonical combined-classroom lesson
• Braid class: C
• Labels: ALL STRANDS launch → 30-1 REQUIRED / 30-2 REQUIRED / 30-3 REQUIRED workshops → CROSS-STRAND EXPOSURE ≠ mastery
• Source: Alberta Mathematics 10-12 Program of Studies, https://open.alberta.ca/publications/mathematics-grades-10-12
• Official outcome IDs today: MATH30-1-TRIGONOM-06-006, MATH30-2-LOGICALR-01-003, MATH30-3-ALGEBRA-01-011

Labels (post on the board)

1. ALL STRANDS - shared launch only.
2. 30-1 REQUIRED - MAT3791 tray; mastery evidence for 30-1 SOs listed below.
3. 30-2 REQUIRED - MAT3792 tray; mastery evidence for 30-2 SOs listed below.
4. 30-3 REQUIRED - MAT3793 tray; mastery evidence for 30-3 SOs listed below.
5. CROSS-STRAND EXPOSURE ≠ mastery - hearing another tray is logged as EXPOSURE, never as an SO tick, never as a silent enrolment change.

ALL STRANDS launch (~10-14 min)

A contamination-test story can feed 30-2 odds and 30-3 interpretation. 30-1 still proves identities.

Clock ~75 min unless the school uses 60; cut the workshop block first, never the strand exit ticket. Class C parallel workshops are the default when one strand is on exclusive work.

Official outcomes (REQUIRED on home strand)

Strand	Official ID	Role today
30-1	MATH30-1-TRIGONOM-06-006	REQUIRED - Prove trigonometric identities, using: • reciprocal identities • quotient identities • Pythagorean identities • sum or difference identities (restricted to sine, cosine and tangent
30-2	MATH30-2-LOGICALR-01-003	REQUIRED - Interpret and assess the validity of odds and probability statements
30-3	MATH30-3-ALGEBRA-01-011	REQUIRED - Analyze and interpret problems that involve probability

30-1 workshop - 30-1 REQUIRED

Clock: ~22-28 min after launch (cut workshop before cutting the strand exit ticket). Learners on this tray stay on 30-1.

MATH30-1-TRIGONOM-06-006

Official text: Prove trigonometric identities, using: • reciprocal identities • quotient identities • Pythagorean identities • sum or difference identities (restricted to sine, cosine and tangent) • double-angle identities (restricted to sine, cosine and tangent)

Worked example (unique to this hour, this strand, this SO): Prove (1 − cos²θ)/sin θ = sin θ for sin θ ≠ 0 using Pythagorean and reciprocal identities. Second item (sum): expand sin(π/7 + θ) using the sum identity (sin, cos, tan only). Third (double-angle): rewrite sin(2θ) if sin θ = 1/4 in QI.

Level	Prompt (inside the home strand)
FOUNDATION	Verify 1+tan²θ=sec²θ at 45°.
CORE	Prove a reciprocal/quotient identity.
PROFICIENT	Sum or double-angle proof with restrictions.
ADVANCED	Reject an illegal step in a near-proof.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Identity map: reciprocal / quotient / Pythagorean / sum-diff / double-angle.

Independent practice: a new instance of today’s SO(s) with different numbers than the worked example. Two sentences on reasonableness.

Wrong-answer repair (do not praise):

• 30-1 SO 006: Moving from one side to the other by adding the same expression without justification, or using sine law as an “identity.” Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.

30-2 workshop - 30-2 REQUIRED

Clock: ~22-28 min after launch (cut workshop before cutting the strand exit ticket). Learners on this tray stay on 30-2.

MATH30-2-LOGICALR-01-003

Official text: Interpret and assess the validity of odds and probability statements

Worked example (unique to this hour, this strand, this SO): A claim: “odds of a false positive are 1:7 so the probability is 1/7.” Convert odds a:b to P=a/(a+b). Assess validity of the original sentence.

Level	Prompt (inside the home strand)
FOUNDATION	Odds 1:1 as a fair coin in words.
CORE	Convert odds to P.
PROFICIENT	Assess a media claim.
ADVANCED	Odds against vs odds in favour.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Odds-versus-probability two-column table.

Independent practice: a new instance of today’s SO(s) with different numbers than the worked example. Two sentences on reasonableness.

Wrong-answer repair (do not praise):

• 30-2 SO 003: Equating odds a:b with probability a/b. Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.

30-3 workshop - 30-3 REQUIRED

Clock: ~22-28 min after launch (cut workshop before cutting the strand exit ticket). Learners on this tray stay on 30-3.

MATH30-3-ALGEBRA-01-011

Official text: Analyze and interpret problems that involve probability

Worked example (unique to this hour, this strand, this SO): A poster: “There is a 90% chance the water is safe.” Critique: 90% of what? Sample size? Given P(fail)=4/100, interpret in words. Do not compute nPr as evidence for this SO.

Level	Prompt (inside the home strand)
FOUNDATION	Likely/unlikely from a given P.
CORE	Interpret a given P in the story.
PROFICIENT	Critique a claim.
ADVANCED	Sample-space talk without nPr.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Probability-statement cards (interpret, do not run nPr).

Independent practice: a new instance of today’s SO(s) with different numbers than the worked example. Two sentences on reasonableness.

Wrong-answer repair (do not praise):

• 30-3 SO 011: Computing nPr or a 30-1 counting proof as if it were 011. Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.

CROSS-STRAND EXPOSURE ≠ mastery

• CROSS-STRAND EXPOSURE ≠ mastery. Log D items. Do not tick another strand’s SO.
• AI Tutor must not treat an exposure card as proficiency.
• 30-3 must not receive 30-1 identity proofs as if they were sine/cosine law.

Project or clean math

Project (Sustainable Community Water & Hygiene) - genuine math only: 30-3: site layout, vehicle if the enterprise needs one, ledger, linear fill-rate, or measurement of volume/chlorine - matching today’s SO.

AI Tutor

• Load the authorized home strand only: MAT3791 (Mathematics 30-1) or MAT3792 (Mathematics 30-2) or MAT3793 (Mathematics 30-3) from the authorized profile.
• Never silent-switch a struggling 30-1 learner into 30-2/30-3 banks (or the reverse) to “make it easier.”
• Serve today’s home-strand SOs. Exposure cards are read-only.
• Never dump diploma keys or released exam wording. Original items only.
• Wrong answers: name the error and repair; do not praise.

Human-tutor handoff

Call a human tutor when: the learner cannot complete CORE after two documented repairs; an identity proof (30-1 006) is stuck after the identity map; a 30-2 017 source is unsafe or empty; a 30-3 vehicle/business number is being faked to match a hoped-for answer; distress, disclosure, or a request to change official enrolment. Enrolment changes are an office/PASI process, not a tutor action.

Diagram / whiteboard

Approved starter math30-quadratic-shift. Strand workshops may add axes/tables. Never praise a translated parabola as a circle.

Strand exit ticket (~8-12 min)

Collect only the home-strand row. Exposure ≠ mastery. Lesson MATH30B-L044.

Home strand	Primary SO	Exit ticket (unique to this lesson)
30-1	MATH30-1-TRIGONOM-06-006	CORE exit: Prove (1 − cos²θ)/sin θ = sin θ for sin θ ≠ 0 using Pythagorean and reciprocal identities. Second item (sum): expand sin(π/7 + θ) using the sum identity (sin, cos, tan only). Third (double-angle): rewrite sin(2θ) if s… FOUNDATION alternate: Verify 1+tan²θ=sec²θ at 45°. Do not submit another strand’s ticket as 30-1 mastery.
30-2	MATH30-2-LOGICALR-01-003	CORE exit: A claim: “odds of a false positive are 1:7 so the probability is 1/7.” Convert odds a:b to P=a/(a+b). Assess validity of the original sentence. FOUNDATION alternate: Odds 1:1 as a fair coin in words. Do not submit another strand’s ticket as 30-2 mastery.
30-3	MATH30-3-ALGEBRA-01-011	CORE exit: A poster: “There is a 90% chance the water is safe.” Critique: 90% of what? Sample size? Given P(fail)=4/100, interpret in words. Do not compute nPr as evidence for this SO. FOUNDATION alternate: Likely/unlikely from a given P. Do not submit another strand’s ticket as 30-3 mastery.

Provenance

Official IDs/text: _official_math_calm_outcomes.json. Pacing: MATH30_COMBINED_CLASSROOM_PACING.md. Crosswalk: MATH30_1_30_2_30_3_OUTCOME_CROSSWALK.csv. Generated by oeom/core/training/rapid3/write_math30b_lessons.py.