Queensland General Mathematics… | Year 11 Maths · T1 W5–6 · Lessons

Continue Weeks 3–4 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, 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. 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, selected A4/text/tactile aids, calculator or paper arithmetic, 20 optional context swaps, and optional local exchange-rate switch. 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.

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.
2–7: 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.
7–13: Learners calculate what 1 additional hour would change and compare $28×10=$280 with $308 to locate the extra premium.
13–20: Routes: move eight regular and two additional hour tokens onto the wage layers aid; 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.
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.

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.

0–2: Ask whether $24 allowance means $24/hour, $24 once or unknown. Source says once for this model shift.
2–7: Model 6×31=$186; 1.25×31=$38.75/hour; 2×38.75=$77.50; then add fixed $24, total $287.50.
7–13: Learners reverse-add the four parts, then estimate 186+about78+24≈288. Contrast an allowance with another hourly rate.
13–20: Routes: use three separate raised trays labelled regular, additional and once-only; annotate the wage layers aid 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.
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.

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.

0–2: Show Card J's $80 stated unit cost and 25% mark-up on cost. Ask which number supplies the percentage base.
2–7: Model $80×0.25=$20 gap and $80+$20=$100 stated selling price; reverse check $100−$80=$20.
7–13: Learners estimate 25% as one quarter of $80 and explain why 25% of $100 would answer a different question.
13–20: Routes: place four equal cost-quarter strips on the base grid; annotate base/gap/price boxes; write or dictate price=cost×1.25 and reverse subtraction. All routes name omitted tax and other costs.
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.

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.

0–2: Read cost $72, price $90, gap $18. Ask why $18 cannot be called a percentage until a denominator is chosen.
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.
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.
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.
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.

Day 25 · Fresh public Check A

Goal: independently transfer layered-pay and percentage-base reasoning to the new festival/print prompt. 25 = 2 + 3 + 12 + 5 + 3.

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.
2–5: Give the exact prompt and neutral access supports without modelling its numbers.
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.
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.
22–25: Consult the public worked key 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.

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.

0–2: Read Card L's two invented list versions: $125 then $137.50. This is not observed inflation.
2–7: Model increase $12.50; 12.50÷125=0.10=10%. Put original $125 under the fraction.
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.
13–20: Routes: place same $12.50 difference against two differently sized base strips on the change ladder; annotate two fraction boxes; calculate and explain both directions orally/typed. Every route identifies the new start and the invented-data boundary.
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.

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.

0–2: Read Card M: in this invented model only, AUD1 = USD0.64 and no fees. Display “not a live quote.”
2–7: Model AUD250 ×0.64 = USD160. Reverse-check USD160 ÷0.64 = AUD250, preserving currency labels.
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.
13–20: Routes: move currency cards through the direction arrows; fill the labelled two-way table; use the optional offline rate switch 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.
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.

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.

0–2: Distinguish annual dividend per share, number of shares and price per share in the invented Paperbird case.
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.
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.
13–20: Routes: group 50 share tokens and one per-share dividend strip on the share measures aid; complete dollars-versus-percent boxes; write/voice per-share and portfolio fractions. All routes name the annual assumption and no investment conclusion.
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.

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.

0–2: Card N2 gives market price and annual earnings per share for Cedar and Slate. Confirm the units match one share.
2–7: Model Cedar $30÷$2=15 and Slate $36÷$3=12. Label both P/E ratios, not 15% and 12%.
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.
13–20: Routes: build price strips from repeated earnings-per-share tokens on the share measures aid; fill a two-company comparison table; write/voice two divisions plus a scope sentence. Optional rate switch is not relevant here; use paper or calculator.
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.

Day 30 · Fresh public Check B

Goal: independently transfer exchange-direction, dividend-yield and P/E reasoning to new fictional data. 25 = 2 + 3 + 12 + 5 + 3.

0–2: Say the case is a public formative check, not a secure exam, exchange quote or financial recommendation. Save first work.
2–5: Give exact prompt, calculator and neutral access supports without demonstrating the new rate or share values.
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.
17–22: After collection, discuss source units, percentage denominator and a missing real condition using different practice values.
22–25: Use the public worked key 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.