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 05 — Exponential growth in the water-enterprise project
Time: 75 minutes
Labels: ALL STRANDS project context · 30-1 logs intro REQUIRED start · 30-2 REQUIRED logs/exponential · 30-3 REQUIRED percent change and cost tables
Braid class: B
Reuse: math30_explog.json 30-1 and 30-2 branches. No 30-3 algebraic log branch.
Dilution as exponential decay: named as later; not assessed today.
Official outcome codes
| Strand | Codes |
|---|---|
| 30-1 | MATH30-1-TRIGONOM-07-013 start (015/016 later) |
| 30-2 | MATH30-2-LOGICALR-04-012; MATH30-2-RELATION-05-013; MATH30-2-RELATION-06-014 REQUIRED start |
| 30-3 | MATH30-3-X-03-007 start (percent, expenses) |
Shared portion (14 min) — ALL STRANDS
Story: weekly users of a refill station: 80, then ×1.15 each week (or a table). Shared sketch. Question: “What is 15% more?” 30-3 owns percent as mastery; 30-1/30-2 use it as context.
30-1 pathway (24 min) — 30-1 REQUIRED start
PATHWAY_30_1 / explog 30-1 branch: definition first.
- \(1.15^t = N\) language; introduce \(\log_{1.15} N\).
- Convert one statement: \(10^x=1000 \iff \log_{10}1000=3\).
| Level | Task |
|---|---|
| FOUNDATION | Rewrite 2^3=8 in log form |
| CORE | log_b a = x iff b^x=a; one exact value |
| PROFICIENT | Why log is the exponent |
| ADVANCED | Change-of-base preview (not required complete) |
30-2 pathway (24 min) — 30-2 REQUIRED
PATHWAY_30_2 / explog 30-2 branch: model + technology after setup.
- Fit language: U(t)=80(1.15)^t. Estimate t when U=200 (tech allowed after writing the model).
- Reasonableness: can users grow 15%/week forever?
| Level | Task |
|---|---|
| FOUNDATION | Compute U(1), U(2) from the formula |
| CORE | Set 80(1.15)^t=200; solve with tech; interpret t |
| PROFICIENT | Two modelling limits |
| ADVANCED | Compare 15%/week vs +12 users/week (linear) — which matches table? |
30-3 pathway (24 min) — 30-3 REQUIRED
- Percent change: 80 → 92 is +15%.
- Cost table: soap, jerrycans, chlorine; weekly expenses vs a simple sales number; profit/loss.
| Level | Task |
|---|---|
| FOUNDATION | Compute one percent increase |
| CORE | Expenses, sales, profit/loss for one week |
| PROFICIENT | Critique: viable if sales drop 20%? |
| ADVANCED | Two-week ledger |
Cross-strand exposure
30-3 does not solve log equations. 30-1 does not critique business viability as 007.
Visuals
VIS-M30-EXPLOG (30-1/30-2) · cost table (30-3) · user-growth sketch ALL STRANDS
Tutor boundaries
Explog adapter: strand from profile 30-1 or 30-2. 30-3: percent/ledger items only. Never serve log laws to 30-3.
Exit tickets
30-1: Write 5^2=25 in log form.
30-2: Model U(t)=80(1.15)^t; find t for U=160 with work shown (tech OK after setup).
30-3: Percent change 80 to 92; profit if sales 400 and expenses 310.
Teacher solutions
30-1 CORE: log_5 25=2.
30-2 CORE: 80(1.15)^t=200 ⇒ (1.15)^t=2.5 ⇒ t=ln(2.5)/ln(1.15)≈6.55 weeks (accept calculator equivalent).
30-2 FOUNDATION: U(1)=92, U(2)=105.8.
30-3 CORE: 15% increase; profit 90.
30-3 PROFICIENT: 20% sales drop → sales 320; if expenses 310, profit 10 — viability weak.
Named later: dilution as exponential decay (chlorine residual) — Session later, not today.
Visual task (BEFORE_LAB): Inspect the visual, notice labels, and answer the lesson prompt.
MATH30B session 05 worked visual 095