# Queensland Year 11 General Mathematics · Unit 1 Topic 1 · Days 21–30

Continue [Weeks 3–4](../weeks-03-04/README.md) after the annual-period, piecework, budget, fully taxable GST, simple-interest and repeated-computation work. This fortnight stays in **General Mathematics 2025 v1.3, Unit 1 Topic 1: Consumer arithmetic**, rather than jumping to Topic 2. The official [QCAA source](https://www.qcaa.qld.edu.au/downloads/senior-qce/syllabuses/snr_maths_general_25_syll.pdf), printed p. 15 / PDF p. 17, includes overtime and allowances, percentage changes, currency exchange, share dividends, P/E ratios and spreadsheet use; see our [row-level coverage and limits](SOURCE-AND-CROSSWALK.md). `GM-U1-T1` is only our local label, **not a QCAA or national code**. This adds ten 25-minute periods, 4 h 10 min; the first 30 model periods total 12 h 30 min, still less than the notional 14-hour Topic 1 subtopic. A school controls actual teaching and formal Unit 1–2 assessment within syllabus rules. These checks are public formative teaching evidence, not those instruments.

**Prepare:** [Cards H–N](LEARNER-CARDS.md), selected [A4/text/tactile aids](print/TEXT-ALTERNATIVES.md), calculator or paper arithmetic, [20 optional context swaps](PRACTICE-SWAPS.md), and optional local [exchange-rate switch](print/rate-switch.xlsx). All wages, prices, rates, shares and companies are fictional. No private data or real employment/investment action is needed. Every route reaches `base / operation / labelled units / reverse or estimate / model limit`. Record content hints separately from reading, AAC or motor supports. A typical 25-minute day uses **2 launch + 5 model + 6 guided + 7 independent + 5 exit/next = 25**. Fresh-check days use **2 + 3 + 12 + 5 + 3 = 25**. Suggested clocks are a plan, not observed classroom timing.

## Day 21 · An overtime multiplier belongs to particular hours

**Goal:** combine regular and additional-hours gross model amounts without multiplying an entire shift by the additional-hours factor. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read Card H: 8 regular hours at $28/hour and 2 additional hours at **1.5× the base hourly rate**. Ask what the 1.5 attaches to.
2. **2–7:** Model `$28×8=$224` regular; new rate `$28×1.5=$42/hour`; additional `$42×2=$84`; gross model total `$308`. Check `224+84` and units. State that 1.5 is a stipulated classroom rule, not an award claim.
3. **7–13:** Learners calculate what 1 additional hour would change and compare `$28×10=$280` with `$308` to locate the extra premium.
4. **13–20:** **Routes:** move eight regular and two additional hour tokens onto the [wage layers aid](print/wage-layers.pdf); fill its two rate/count rows in large text; write/voice `8×28+2×(1.5×28)` with units. All routes identify one missing real-work condition.
5. **20–25:** Exit “Why is `10×42` wrong?” **Key:** only two hours have the stated additional rate. **Move:** circle the count linked to each rate before recalculating.

**Alternative domains:** fictional film crew or costume repair. **Optional/home:** make up 4 regular hours at $20 and 1 additional hour at 1.5×; no real timesheet. [Two optional swaps](PRACTICE-SWAPS.md#day-21--two-bands).

## Day 22 · An allowance is a separate unit

**Goal:** add Card I's one-shift fixed allowance after calculating regular and additional-hour amounts. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Ask whether `$24 allowance` means `$24/hour`, `$24 once` or unknown. Source says once for this model shift.
2. **2–7:** Model `6×31=$186`; `1.25×31=$38.75/hour`; `2×38.75=$77.50`; then add fixed `$24`, total `$287.50`.
3. **7–13:** Learners reverse-add the four parts, then estimate `186+about78+24≈288`. Contrast an allowance with another hourly rate.
4. **13–20:** **Routes:** use three separate raised trays labelled regular, additional and once-only; annotate the [wage layers aid](print/wage-layers.pdf) with a third fixed layer; enter/speak an expression `6×31+2×(1.25×31)+24`, labelling each term. Every route rejects a real entitlement claim.
5. **20–25:** Exit “If the $24 is added for every hour, what was misread?” **Key:** allowance unit was one shift. **Move:** rewrite each input as `dollars per hour`, `hours`, or `dollars once` before operations.

**Alternative domains:** invented gallery install or recording studio support. **Optional/home:** invent a one-time $10 materials allowance beside 3 hours at $18/hour, then say which conditions remain unknown. [Two optional swaps](PRACTICE-SWAPS.md#day-22--allowance-once).

## Day 23 · Mark-up begins with cost

**Goal:** use the cost base to calculate a fictional mark-up and selling price. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Show Card J's $80 stated unit cost and 25% **mark-up on cost**. Ask which number supplies the percentage base.
2. **2–7:** Model `$80×0.25=$20` gap and `$80+$20=$100` stated selling price; reverse check `$100−$80=$20`.
3. **7–13:** Learners estimate 25% as one quarter of $80 and explain why `25% of $100` would answer a different question.
4. **13–20:** **Routes:** place four equal cost-quarter strips on the [base grid](print/markup-margin.pdf); annotate base/gap/price boxes; write or dictate `price=cost×1.25` and reverse subtraction. All routes name omitted tax and other costs.
5. **20–25:** Exit “Is the $20 proven net profit?” **Key:** no; extra costs are omitted. **Move:** separate mathematical unit price gap from a business net-profit conclusion.

**Alternative domains:** fictional game-event badge or community theatre prop. **Optional/home:** model a made-up $40 base with 10% mark-up; no purchase. [Two optional swaps](PRACTICE-SWAPS.md#day-23--cost-base).

## Day 24 · One dollar gap, two percentages

**Goal:** calculate mark-up on cost and margin on selling price for Card K, with both denominators visible. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read cost $72, price $90, gap $18. Ask why $18 cannot be called a percentage until a denominator is chosen.
2. **2–7:** Model mark-up `18÷72×100=25%`, then margin `18÷90×100=20%`. In each case, mark the denominator on the [base grid](print/markup-margin.pdf).
3. **7–13:** Learners reconstruct price from cost plus gap and explain in one sentence why both percentages can be correct here. This is a gross gap, not net margin after expenses.
4. **13–20:** **Routes:** lay the $18 gap strip beside $72 then $90 labelled bases; fill a two-row ratio table; give a spoken/typed explanation with two fractions and units. All routes distinguish denominator and business limit.
5. **20–25:** Exit “Which denominator belongs to margin?” **Key:** selling price, $90. **Move:** ask learner to point to what the named percentage is “of.”

**Alternative domains:** invented app-icon printing or repair-shop parts. **Optional/home:** with fictional cost $20 and selling price $25, find both percentages; no real shop data. [Two optional swaps](PRACTICE-SWAPS.md#day-24--two-bases).

## Day 25 · Fresh public Check A

**Goal:** independently transfer layered-pay and percentage-base reasoning to the new [festival/print prompt](STUDENT-CHECKS.md#check-a--day-25--payment-layers-and-two-percentage-bases). **25 = 2 + 3 + 12 + 5 + 3.**

1. **0–2:** Name the check public, fresh relative to cards and **not a secure examination** or QCAA instrument. Explain first work will be kept before feedback.
2. **2–5:** Give the exact prompt and neutral access supports without modelling its numbers.
3. **5–17:** **Routes:** arrange labelled regular/additional/allowance and cost/price strips; complete blank two-base tables; type, write, dictate or AAC the same calculations and reasons. Keep the first independent response and note any mathematical hint.
4. **17–22:** After collection, ask “which count meets which rate?” and “which denominator names mark-up or margin?” Use a different invented example, never erase first evidence.
5. **22–25:** Consult the [public worked key](teacher/KEY-AND-NEXT.md) to choose one response move. **Move:** if a learner already saw answers, use a new local parallel check.

**Alternative domains:** fictional convention booth or sports-event crew only **after** first collection. **Optional/home:** explain why a fixed allowance and hourly rate use different units. [Two after-check swaps](PRACTICE-SWAPS.md#day-25--after-check-only).

## Day 26 · Percentage change needs its own starting base

**Goal:** find a forward increase and return decrease without assuming equal percentages. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read Card L's two invented list versions: $125 then $137.50. This is not observed inflation.
2. **2–7:** Model increase `$12.50`; `12.50÷125=0.10=10%`. Put **original $125** under the fraction.
3. **7–13:** Return direction uses `$12.50÷137.50≈9.09%` decrease. Learners show why a 10% decrease from $137.50 would not land exactly on $125.
4. **13–20:** **Routes:** place same $12.50 difference against two differently sized base strips on the [change ladder](print/change-ladder.pdf); annotate two fraction boxes; calculate and explain both directions orally/typed. Every route identifies the new start and the invented-data boundary.
5. **20–25:** Exit “Same dollar change: same percentage?” **Key:** no, because the starting bases differ. **Move:** underline “from” in each sentence before choosing a denominator.

**Alternative domains:** fictional festival supply list or book-cover estimate. **Optional/home:** invent a $50→$60 change and reverse it; no actual inflation claim. [Two optional swaps](PRACTICE-SWAPS.md#day-26--forward-and-return).

## Day 27 · Exchange direction changes the operation

**Goal:** apply a stipulated AUD→USD rate and invert it when converting back. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read Card M: in this **invented model only**, AUD1 = USD0.64 and no fees. Display “not a live quote.”
2. **2–7:** Model AUD250 ×0.64 = USD160. Reverse-check USD160 ÷0.64 = AUD250, preserving currency labels.
3. **7–13:** Learners separately convert USD96 ÷0.64 = AUD150 and reverse ×0.64 = USD96. An estimate notes AUD amount should be numerically larger here because each AUD maps to less than one USD in this model.
4. **13–20:** **Routes:** move currency cards through the [direction arrows](print/exchange-directions.pdf); fill the labelled two-way table; use the optional [offline rate switch](print/rate-switch.xlsx) or write/speak both expressions and check them by hand. Every route states rate date/fees are unknown in reality; no live rate is fetched.
5. **20–25:** Exit “USD96 ×0.64 gives what, and why is it not the requested AUD amount?” **Key:** it applies the AUD→USD direction to the wrong starting currency. **Move:** write the source unit above each number, then choose multiplication/division.

**Alternative domains:** invented travel itinerary or digital licensing estimate. **Optional/home:** change a *made-up* conversion factor and predict direction; do not look up or use a live rate. [Two optional swaps](PRACTICE-SWAPS.md#day-27--stipulated-rates).

## Day 28 · Dividend dollars and yield answer different questions

**Goal:** compute Card N1's total annual dividend model and per-share yield with correctly named bases. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Distinguish annual dividend **per share**, number of shares and price **per share** in the invented Paperbird case.
2. **2–7:** Model `50×$0.84=$42` annual dividend dollars; yield `$0.84÷$24×100=3.5%`. The 50-share count affects total dollars, not yield percentage per share.
3. **7–13:** Learners use `50×$24=$1,200` fictional portfolio price and check `$42÷$1,200×100=3.5%`. Say why this is an arithmetic equivalence, not a future return guarantee.
4. **13–20:** **Routes:** group 50 share tokens and one per-share dividend strip on the [share measures aid](print/share-measures.pdf); complete dollars-versus-percent boxes; write/voice per-share and portfolio fractions. All routes name the annual assumption and no investment conclusion.
5. **20–25:** Exit “Why not divide $42 by $24 for yield?” **Key:** that mixes portfolio dividend with one-share price. **Move:** align numerator and denominator to one share or the full 50-share portfolio.

**Alternative domains:** invented co-op simulation or fictional textbook publisher. **Optional/home:** calculate an imagined 10 shares × $0.20 annual dividend; no actual share research. [Two optional swaps](PRACTICE-SWAPS.md#day-28--dividend-and-yield).

## Day 29 · P/E is a ratio, not a recommendation

**Goal:** calculate two fictional price-to-earnings ratios and reject an unsupported investment ranking. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Card N2 gives market price and **annual earnings per share** for Cedar and Slate. Confirm the units match one share.
2. **2–7:** Model Cedar `$30÷$2=15` and Slate `$36÷$3=12`. Label both P/E **ratios**, not 15% and 12%.
3. **7–13:** Learners reverse-check `15×$2=$30`, `12×$3=$36`; list absent growth, debt, risk, accounting quality and time context. Lower P/E alone cannot select a share.
4. **13–20:** **Routes:** build price strips from repeated earnings-per-share tokens on the [share measures aid](print/share-measures.pdf); fill a two-company comparison table; write/voice two divisions plus a scope sentence. Optional [rate switch](print/rate-switch.xlsx) is **not** relevant here; use paper or calculator.
5. **20–25:** Exit “Can we call Slate the better investment because 12<15?” **Key:** no; ratio alone is insufficient. **Move:** ask for one missing decision input before any comparative claim.

**Alternative domains:** wholly fictional publishing or game-studio company cards. **Optional/home:** make a ratio for invented price $20 and earnings/share $2; no real-market lookup. [Two optional swaps](PRACTICE-SWAPS.md#day-29--price-to-earnings).

## Day 30 · Fresh public Check B

**Goal:** independently transfer exchange-direction, dividend-yield and P/E reasoning to [new fictional data](STUDENT-CHECKS.md#check-b--day-30--direction-dividend-and-ratio). **25 = 2 + 3 + 12 + 5 + 3.**

1. **0–2:** Say the case is a **public formative check**, not a secure exam, exchange quote or financial recommendation. Save first work.
2. **2–5:** Give exact prompt, calculator and neutral access supports without demonstrating the new rate or share values.
3. **5–17:** **Routes:** move AUD/USD direction and per-share/portfolio cards; fill blank labelled operation boxes; type, write, dictate or AAC the same calculations with units and limits. Record mathematical hints separately.
4. **17–22:** After collection, discuss source units, percentage denominator and a missing real condition using different practice values.
5. **22–25:** Use the [public worked key](teacher/KEY-AND-NEXT.md) to choose a targeted next lesson. **Move:** if public prior access affects independence, set a new locally authored prompt.

**Alternative domains:** invented ticket-credit conversion and game-company metrics **after** first collection. **Optional/home:** write a one-sentence reason why neither a made-up exchange rate nor a single P/E can support a real transaction. [Two after-check swaps](PRACTICE-SWAPS.md#day-30--after-check-only).