Canonical combined lesson MATH30B-L069 · week 14 day 4 · CURRICULUM_BASELINE

MATH30B-L069 - Mixed combinatorics; rationals review; 007-008 fill

• Delivery id: MATH30B (combined classroom; not a fourth credential)
• Official courses: Mathematics 30-1 (MAT3791), 30-2 (MAT3792), 30-3 (MAT3793)
• Lesson ID: MATH30B-L069
• Calendar: Week 14 Day 4 of 18 weeks × 5 days (lesson 69 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-01-021, MATH30-1-TRIGONOM-02-022, MATH30-1-TRIGONOM-03-023, MATH30-1-TRIGONOM-04-024, MATH30-2-LOGICALR-01-009, MATH30-2-LOGICALR-02-010, MATH30-2-LOGICALR-03-011, MATH30-3-X-03-007, MATH30-3-ALGEBRA-01-008

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)

Teacher-written mixed. Strand trays stay labelled REQUIRED.

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-01-021	REQUIRED - Apply the fundamental counting principle to solve problems
30-1	MATH30-1-TRIGONOM-02-022	REQUIRED - Determine the number of permutations of n elements taken r at a time to solve problems
30-1	MATH30-1-TRIGONOM-03-023	REQUIRED - Determine the number of combinations of n different elements taken r at a time to solve problems
30-1	MATH30-1-TRIGONOM-04-024	REQUIRED - Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers)
30-2	MATH30-2-LOGICALR-01-009	REQUIRED - Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials and binomials)
30-2	MATH30-2-LOGICALR-02-010	REQUIRED - Perform operations on rational expressions (limited to numerators and denominators that are monomials and binomials)
30-2	MATH30-2-LOGICALR-03-011	REQUIRED - Solve problems that involve rational equations (limited to numerators and denominators that are monomials and binomials)
30-3	MATH30-3-X-03-007	REQUIRED - Critique the viability of small business options by considering: • expenses • sales • profit or loss
30-3	MATH30-3-ALGEBRA-01-008	REQUIRED - Demonstrate an understanding of linear relations by: • recognizing patterns and trends • graphing • creating tables of values • writing equations • interpolating and extrapolating

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-01-021

Official text: Apply the fundamental counting principle to solve problems

Worked example (unique to this hour, this strand, this SO): A lock uses 8 digits (0-9) then 7 letters (26). Count codes if repetition is allowed, then if digits cannot repeat. Add a restriction: first digit not 0.

Level	Prompt (inside the home strand)
FOUNDATION	Two-stage menu count.
CORE	Multi-stage FCP.
PROFICIENT	FCP with a restriction.
ADVANCED	Justify an overcount.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Tree diagram VIS-M30B-FCP.

MATH30-1-TRIGONOM-02-022

Official text: Determine the number of permutations of n elements taken r at a time to solve problems

Worked example (unique to this hour, this strand, this SO): Compute P(14,3) = n!/(n−r)!. Schedule 3 distinct speakers from 14 where order of speaking matters. Contrast a case with a repeated identical role (preview only if ADVANCED).

Level	Prompt (inside the home strand)
FOUNDATION	List permutations of 3 letters.
CORE	P(n,r) formula.
PROFICIENT	Restrictions (cannot be first).
ADVANCED	Identical items preview.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Ordered-arrangement visual VIS-M30B-PERM.

MATH30-1-TRIGONOM-03-023

Official text: Determine the number of combinations of n different elements taken r at a time to solve problems

Worked example (unique to this hour, this strand, this SO): Compute C(14,3). Choose 3 test kits from 14 where order does not matter. Show C(n,r)=P(n,r)/r! and why dividing by r! is required.

Level	Prompt (inside the home strand)
FOUNDATION	List combinations of 3 from 4.
CORE	C(n,r) formula.
PROFICIENT	Choose nCr vs nPr.
ADVANCED	Restriction on a committee.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Unordered-subset visual (nCr) distinct from nPr.

MATH30-1-TRIGONOM-04-024

Official text: Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers)

Worked example (unique to this hour, this strand, this SO): Expand (x+8)^9 using the binomial theorem (natural exponent). Give the general term and the term containing x^7.

Level	Prompt (inside the home strand)
FOUNDATION	(x+1)² by Pascal.
CORE	Full expansion natural n.
PROFICIENT	General term.
ADVANCED	Coefficient of a named power.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Pascal row / binomial expansion aligned to coefficients.

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 021: Adding choices instead of multiplying stages. Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.
• 30-1 SO 022: Using nCr when order matters. Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.
• 30-1 SO 023: Using nPr when order does not matter. Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.
• 30-1 SO 024: Writing binomial coefficients as n+r or expanding with subtraction signs ignored. 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-01-009

Official text: Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials and binomials)

Worked example (unique to this hour, this strand, this SO): Simplify (x²−64)/(x−8) for x≠8. State the equivalent form and the restriction. Numerators/denominators limited to monomials and binomials.

Level	Prompt (inside the home strand)
FOUNDATION	Cancel a monomial factor.
CORE	Binomial over binomial with restriction.
PROFICIENT	State all non-permissible values.
ADVANCED	Equivalent form used later in an equation.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Expression tiles showing cancelled factors and restrictions.

MATH30-2-LOGICALR-02-010

Official text: Perform operations on rational expressions (limited to numerators and denominators that are monomials and binomials)

Worked example (unique to this hour, this strand, this SO): Compute (x/8) + (7/(x+2)) with LCD x(x+2), x≠0, x≠−2. Then multiply (x/7)·((x+8)/x) and state restrictions.

Level	Prompt (inside the home strand)
FOUNDATION	Multiply two monomial rationals.
CORE	Add with LCD (binomials).
PROFICIENT	Complex fraction light.
ADVANCED	All restrictions after operations.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Common-denominator layout for rational operations.

MATH30-2-LOGICALR-03-011

Official text: Solve problems that involve rational equations (limited to numerators and denominators that are monomials and binomials)

Worked example (unique to this hour, this strand, this SO): Solve 8/x + 7/(x+2) = 7/(x(x+2)) with x≠0, x≠−2. Check for extraneous roots. Context option: two work rates filling a tank.

Level	Prompt (inside the home strand)
FOUNDATION	Clear a single-term rational equation.
CORE	Binomial denominators + check.
PROFICIENT	Work-rate context.
ADVANCED	Explain an extraneous root.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Rate table that becomes a rational equation; restrictions boxed.

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 009: Cancelling terms (not factors), or dropping restrictions. Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.
• 30-2 SO 010: Adding numerators and denominators separately. Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.
• 30-2 SO 011: Clearing denominators then keeping a restricted x as a root. 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-X-03-007

Official text: Critique the viability of small business options by considering: • expenses • sales • profit or loss

Worked example (unique to this hour, this strand, this SO): Month expenses $2280, sales $2825, so profit/loss = sales − expenses. Second scenario: sales drop 16%. Critique viability (can this hygiene kiosk continue?).

Level	Prompt (inside the home strand)
FOUNDATION	Label expense vs revenue.
CORE	Compute profit or loss.
PROFICIENT	Critique viability.
ADVANCED	Two scenarios.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. Ledger: expenses, sales, profit or loss.

MATH30-3-ALGEBRA-01-008

Official text: Demonstrate an understanding of linear relations by: • recognizing patterns and trends • graphing • creating tables of values • writing equations • interpolating and extrapolating • solving problems

Worked example (unique to this hour, this strand, this SO): Table: hours 0, 1, 2, 3 vs litres treated 0, 20, 40, 60. Graph, write litres = (rate)(hours), interpolate at 1.5 h, extrapolate to 5 h with a caution.

Level	Prompt (inside the home strand)
FOUNDATION	Complete a table.
CORE	Graph + equation.
PROFICIENT	Interpolate.
ADVANCED	Extrapolate with caution.

Diagram: Timing: after retrieval, before independent practice. Do not open the diagram during the first retrieval prompt. VIS-M30B-LINEAR: table, graph, equation together.

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 007: Calling revenue “profit,” or ignoring a category of expense. Repair: name the first wrong line, replace it, and re-complete the CORE tray item. Do not praise the wrong line.
• 30-3 SO 008: Extrapolating far beyond the table with no caution. 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.

Project or clean math

Project overlay today: none. Use a clean math prompt on the home-strand tray. Do not force Sustainable Community Water & Hygiene onto work that is not genuinely that mathematics.

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-L069.

Home strand	Primary SO	Exit ticket (unique to this lesson)
30-1	MATH30-1-TRIGONOM-01-021	CORE exit: A lock uses 8 digits (0-9) then 7 letters (26). Count codes if repetition is allowed, then if digits cannot repeat. Add a restriction: first digit not 0. FOUNDATION alternate: Two-stage menu count. Do not submit another strand’s ticket as 30-1 mastery.
30-2	MATH30-2-LOGICALR-01-009	CORE exit: Simplify (x²−64)/(x−8) for x≠8. State the equivalent form and the restriction. Numerators/denominators limited to monomials and binomials. FOUNDATION alternate: Cancel a monomial factor. Do not submit another strand’s ticket as 30-2 mastery.
30-3	MATH30-3-X-03-007	CORE exit: Month expenses $2280, sales $2825, so profit/loss = sales − expenses. Second scenario: sales drop 16%. Critique viability (can this hygiene kiosk continue?). FOUNDATION alternate: Label expense vs revenue. 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.