Mathematics 30 (braided classroom)
One combined classroom for Mathematics 30-1, 30-2 and 30-3. Braiding adjusts the shared calendar. It does not shrink any strand to ten sessions.
MATH30B Session 04 — Inverse and composition intro
Time: 75 minutes
Labels: 30-1 REQUIRED · CROSS-STRAND EXPOSURE for 30-2 and for 30-3 inverse-as-undo language · 30-3 REQUIRED continues dilations (004)
Braid class: E (30-1 007/012) · D (30-2/30-3 inverse) · C (30-3 scale drawings)
Official outcome codes
| Strand | Codes |
|---|---|
| 30-1 | MATH30-1-TRIGONOM-01-007; MATH30-1-TRIGONOM-06-012 REQUIRED |
| 30-2 | none for mastery |
| 30-3 | MATH30-3-X-03-004 (dilation as inverse of enlargement) |
Shared portion (10 min) — ALL STRANDS
Demo: a scale drawing doubled, then halved. “Undo.” Demo: a number 3→7 via 2x+1, then undo. Tell the room: only 30-1 is assessed on inverse of relations today.
30-1 pathway (42 min) — 30-1 REQUIRED
PATHWAY_30_1: algebraic inverse, domain, composition.
- f(x)=2x+1. Find f^{-1}. Restrict if needed for later non-linear examples (√x).
- Numeric composition (f∘g)(x); one (f∘f^{-1})(x)=x check on domain.
| Level | Task |
|---|---|
| FOUNDATION | Undo 2x+1 numerically; swap coordinates of three points |
| CORE | Algebraic inverse of a linear f; state domain/range swap |
| PROFICIENT | (f∘f^{-1})(x)=x on the correct domain |
| ADVANCED | Why y=x² (unrestricted) fails to have an inverse function |
30-2 pathway (42 min) — CROSS-STRAND EXPOSURE
Graphic undo: reflect a scatter through y=x as a picture. One sentence: “this undoes the input-output visually.” Mark EXPOSURE. PATHWAY_30_2 later uses graphic inverses when algebraic inverse is opaque — not a 30-2 SO.
| Level | Task (all ungraded as 012) |
|---|---|
| FOUNDATION | Trace three points to reflected points |
| CORE | Sentence: output becomes input |
| PROFICIENT | When the picture is not a function after reflection |
| ADVANCED | Optional listen to 30-1 domain talk — still EXPOSURE |
30-3 pathway (42 min) — 30-3 REQUIRED
Site plan of a wash-station. Enlarge by k=3; then apply k=1/3. State: dilation by 1/k undoes dilation by k (from the same centre).
| Level | Task |
|---|---|
| FOUNDATION | Enlarge a rectangle by 2 |
| CORE | Enlarge then reduce; original recovered |
| PROFICIENT | Scale drawing: 1 cm : 1.5 m; convert one length |
| ADVANCED | Two centres — why inverse fails if centre changes |
Cross-strand exposure
D-S04-INV, D-S04-COMP. Log for 30-2 and 30-3. Never write 012 or 007 on their mastery maps.
Visuals
VIS-M30B-INV (y=x) · scale-drawing grid · mapping diagram for composition
Tutor boundaries
30-1: inverse/composition algebra.
30-2: exposure card only (ungraded).
30-3: dilation/scale items.
Refuse 30-2 log or permutation items today.
Exit tickets
30-1: Inverse of f(x)=3x-6 and one composition check.
30-2: EXPOSURE: sketch the reflection of two points through y=x; label EXPOSURE.
30-3: Scale factor 4 then 1/4 on a segment; lengths before/after.
Teacher solutions
30-1 CORE: f(x)=3x-6 ⇒ y=3x-6 ⇒ x=3y-6 ⇒ f^{-1}(x)=(x+6)/3.
30-1 PROFICIENT: (f∘f^{-1})(x)=3·((x+6)/3)-6=x.
30-1 ADVANCED: y=x² fails horizontal line test on ℝ; restrict x≥0 to invert with √.
30-2: Points (1,2)→(2,1), (0,4)→(4,0).
30-3 CORE: Length 5 cm → 20 cm → 5 cm if k=4 then 1/4, same centre.
Misconception: “Everyone learned inverses, so everyone has the inverse outcome.” False.
Visual task (DURING_WORKED_EXAMPLE): Inspect the visual, notice labels, and answer the lesson prompt.
MATH30B session 04 comparison matrix 094