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 02 — Representing change
Time: 75 minutes
Labels: ALL STRANDS shared graphs · 30-1 REQUIRED function notation/domain · 30-2 REQUIRED data representation · 30-3 REQUIRED linear relations / measurement precision
Braid class: B
AI Tutor: home-strand items only; authorized profile change to switch.
Official outcome codes
| Strand | Codes |
|---|---|
| 30-1 | MATH30-1-TRIGONOM-01-007 (start: operations language via notation; composition preview light) |
| 30-2 | MATH30-2-RELATION-06-014 (start: represent data — tables/graphs; exp/log fit not required today) |
| 30-3 | MATH30-3-ALGEBRA-01-008; MATH30-3-X-01-001 |
Shared portion (15 min) — ALL STRANDS
Context: litres of treated water vs day for a small hygiene enterprise (table of 6 days). Shared questions: What is changing? What stays still? Sketch axes together. Do not declare one model (linear vs exponential) as the class answer.
30-1 pathway (22 min) — 30-1 REQUIRED
PATHWAY_30_1: notation and domain before story.
- Define \(W(d)\) = water output on day \(d\).
- Compute \(W(4)\) from a piecewise or formula the teacher provides, e.g. \(W(d)=120+15d\) on \(d=0,1,\ldots,10\).
- Domain: why \(d=-1\) is rejected; why \(d=3.5\) may or may not be (discrete days vs continuous model).
| Level | Task |
|---|---|
| FOUNDATION | Evaluate W at two integers from a table |
| CORE | Write W(4) from formula; domain sentence |
| PROFICIENT | If V(d)=2W(d), interpret V as a scalar multiple (operation on functions — start of 007) |
| ADVANCED | Numeric composition: cost C(W)=0.02W; compute (C∘W)(4) and name inner/outer |
30-2 pathway (22 min) — 30-2 REQUIRED
PATHWAY_30_2: data first.
- Plot the six points. Describe trend in words.
- Decide: does a straight line look reasonable? What would make you suspect faster-than-linear growth? (Sets up S05; do not force log today.)
| Level | Task |
|---|---|
| FOUNDATION | Plot; say increasing/decreasing |
| CORE | Describe rate in words using the table |
| PROFICIENT | Sketch a possible curve; list two modelling assumptions |
| ADVANCED | First differences vs rough ratios — which is stabler? |
30-3 pathway (22 min) — 30-3 REQUIRED
- Build table of values; graph; write a linear equation that fits *this* table if it is linear (teacher uses a linear data set for 30-3 even if 30-2 sees a different overlay — or same numbers if they are linear; if shared table is nonlinear, 30-3 uses a linear subset labelled 30-3 REQUIRED).
- Interpolate day 2.5 if the model is continuous; state caution.
- Precision: each litre reading has instrument uncertainty (001).
| Level | Task |
|---|---|
| FOUNDATION | Complete table; plot points |
| CORE | Equation W=120+15d (example); interpolate one value |
| PROFICIENT | Extrapolate day 12 with a caution sentence |
| ADVANCED | How ±0.5 L uncertainty moves the graph |
Cross-strand exposure
30-1 hears 30-3 precision language (D). Not 001 mastery.
30-3 hears f(d) spoken. Not 007.
Visuals
VIS-M30B-DATA (scatter) · VIS-M30B-LINEAR (30-3) · mapping diagram (30-1)
Tutor boundaries
30-1: notation/domain items. 30-2: describe-data items. 30-3: table/graph/equation/interpolate + uncertainty. No explog solve yet.
Exit tickets
30-1: W(3) and domain of W.
30-2: Two sentences: what the graph shows; one assumption.
30-3: Equation of the linear model + one interpolated value + uncertainty of the instrument.
Teacher solutions (example data \(W=120+15d\), d=0..5)
- W(4)=180 L.
- 30-1 PROFICIENT: V(d)=2W(d)=240+30d.
- 30-1 ADVANCED: (C∘W)(4)=0.02×180=3.60 (currency). Inner W, outer C.
- 30-2 CORE: increases by 15 L per day (constant first differences).
- 30-3 CORE: W=120+15d; interpolate d=2.5 → 157.5 L; if instrument ±0.5 L, report interval.
- 30-3 PROFICIENT: day 12 → 300 L only if linearity and operations hold.
Misconception: Treating a discrete day-count as if d can be −3 without comment.
Visual task (DURING_DIRECT_INSTRUCTION): Inspect the visual, notice labels, and answer the lesson prompt.
MATH30B session 02 strand map 092