# Queensland Year 11 General Mathematics · Unit 1 Topic 1 · ten 25-minute teacher scripts

This continues the [opening mathematics fortnight](../../../term-1/weeks-01-02/mathematics/LESSONS.md): rates, percentage changes, unit price and a small planning budget. Days 11–20 extend consumer arithmetic into salary-period conversion, piecework and commission, a category-aware budget, fully taxable GST exercises, simple interest and a repeated computation table. The exact current QCAA source is [General Mathematics 2025 v1.3](https://www.qcaa.qld.edu.au/downloads/senior-qce/syllabuses/snr_maths_general_25_syll.pdf), Unit 1 **Money, measurement, algebra and linear equations**, Topic 1 **Consumer arithmetic**, PDF p. 17 / printed p. 15; see [source/coverage crosswalk](SOURCE-AND-CROSSWALK.md). `GM-U1-T1` is only SubjectNest shorthand, not an official QCAA code. Ten 25-minute periods are **4 hours 10 minutes**, so this continuation cannot complete the syllabus's 14-hour Topic 1 or 55-hour Unit 1. Schools decide sequencing and formal Unit 1–2 assessment. These public checks are teaching observations, not QCAA instruments.

**Prepare:** [Cards A–G](LEARNER-CARDS.md), five [A4/text/tactile aids](print/TEXT-ALTERNATIVES.md), calculator or paper arithmetic, optional [local what-if spreadsheet](print/budget-what-if.xlsx). All financial inputs are invented; nobody shares personal money or labour data. Ask learners to show `given / operation / units / reasonableness check / limitation`. A route changes representation, not the expected mathematical reasoning or a learner identity. Record a substantive hint separately from reading, AAC or motor access. On an ordinary day the clock is **2 launch + 5 explicit model + 6 guided practice + 7 independent choice + 5 exit/response = 25 minutes**. Check days use **2 + 3 + 12 + 5 + 3 = 25**.

## Day 11 · One annual amount, three period averages

**Goal:** convert Card A's gross annual model amount to correctly labelled weekly, fortnightly and monthly averages. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Display `$65,520 per year`; ask why `$65,520 ÷ 12` and `÷52` answer different questions. Read “gross” and “model” aloud.
2. **2–7:** Model `$65,520 ÷ 52 = $1,260 per model week`; multiply by two for `$2,520 per fortnight`. Check `$1,260×52=$65,520`. Do not call this net take-home pay.
3. **7–13:** Learners find monthly `$65,520÷12=$5,460`, verify `5,460×12`, and contrast average month with unequal real calendar month lengths.
4. **13–20:** **Routes:** move annual→52/26/12 period cards and write equations; complete the [period table](print/period-converter.pdf) in large print; type or dictate three expressions with units and a reverse-multiplication check. All routes name gross-versus-net uncertainty.
5. **20–25:** Exit “Which divisor gives a fortnightly model average?” **Key:** 26, yielding $2,520. **Move:** if a learner multiplies by 26, use `26 periods × amount/period = annual amount` to restore direction.

**Alternative domain:** A fictional theatre production salary, not a real contract. **Optional/home:** convert an invented $52,000 annual model; no personal payslip.

## Day 12 · Paid time versus accepted pieces

**Goal:** compute two stipulated gross payment models without treating one as a real legal choice. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read Card B's units: dollars per paid hour and dollars per accepted prop. Ask which count belongs to each rate.
2. **2–7:** Model hourly `5×$32=$160`; piecework `20×$9=$180`. Check `20×10≈200`, so $180 is plausible. The $20 difference is for these stated counts only.
3. **7–13:** Learners compute 15 accepted props = $135 and compare with $160; discuss why fewer accepted props changes only the piecework side in this exercise.
4. **13–20:** **Routes:** pair hour and prop count tiles with their rates; annotate a two-column [payment board](print/payment-models.pdf); write/speak two unit-cancelling multiplication chains. Every route gives both totals, comparison and one unknown work condition.
5. **20–25:** Exit “Does $180 prove piecework is a better real job?” **Key:** no; hours, conditions, safety, legal pay entitlements and deductions are absent. **Move:** if a learner compares $32 with $9 alone, restore the required counts and units.

**Alternative domain:** Invented maker-fair badge assembly. **Optional/home:** compare 3 paid hours at a made-up $30/h with 12 accepted pieces at $8 each; no work undertaken.

## Day 13 · Whole-number threshold in piecework

**Goal:** find and justify the first whole item count that exceeds a fixed stated total. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Retrieve Card B's $160 hourly total and $9 per accepted prop. Ask “How many accepted props first give **more than** $160?”
2. **2–7:** Model `9n > 160`; `n > 160/9≈17.78`, so the first whole count is **18**, giving $162. This is a mathematical boundary, not a productivity target.
3. **7–13:** Learners check 17 props = $153 and 18 = $162. Contrast “at least $160” with “strictly more than $160”; the threshold happens to be 18 for both here, so ask why the words still matter.
4. **13–20:** **Routes:** build a $9-per-piece number line using tokens; fill 15–20 rows in the payment board; solve the inequality and verify adjacent whole counts. All routes show the two boundary values and the unknown real work time.
5. **20–25:** Exit “Why not round 17.78 to 17?” **Key:** 17 gives only $153. **Move:** make learner test both adjacent whole counts before selecting a conclusion.

**Alternative domain:** Fictional photo-editing deliverables. **Optional/home:** find the first whole count at $7 each exceeding a fictional $50 target; no real gig advice.

## Day 14 · Commission is a percentage of a named base

**Goal:** calculate Card C base-plus-commission scenarios and interpret percentage base. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Point to “4.5% of eligible sales”. Ask whether 4.5% is taken from $180 base, $1,200 sales or total pay.
2. **2–7:** Model `0.045×$1,200=$54` commission; `180+54=$234` gross model shift total. Estimate 5% of $1,200 is $60, so $54 is credible.
3. **7–13:** Learners compute $800 sales → $36 commission/$216 total and $1,600 → $72/$252; compare how an extra $400 eligible sales adds $18 in this fixed rule.
4. **13–20:** **Routes:** move a base-pay block plus 4.5%-of-sales strips; complete the [commission table](print/payment-models.pdf); write a formula `T(s)=180+0.045s` and explain it orally/AAC. All routes label base, sales, commission and unknown real conditions.
5. **20–25:** Exit “Why is $180 + 4.5 = $184.50 wrong?” **Key:** 4.5 is a percent, not a dollar commission. **Move:** estimate 5% of $1,200 first, then calculate exactly.

**Alternative domain:** Imaginary adult gallery sales desk. **Optional/home:** calculate 3% of invented $600 eligible sales with a fictional $200 base; no actual sales.

## Day 15 · Fresh public Check A

**Goal:** independently transfer period conversion and commission-base reasoning to [new fictional data](STUDENT-CHECKS.md#check-a--day-15--different-payment-periods). **25 = 2 + 3 + 12 + 5 + 3.**

1. **0–2:** Explain this public prompt is fresh relative to lessons, **not a secure exam** or QCAA assessment. Adapt locally if previously seen.
2. **2–5:** Give exact prompt, large text, calculator, tactile period cards, AAC/sign or exact-word scribe without demonstrating its new numbers.
3. **5–17:** Learners answer independently; retain first work before feedback. Note access, interpretive cue and calculation separately.
4. **17–22:** After collection, reflect on `period / percent base / unit / missing condition`. **Routes:** period-card reconstruction; table annotation; spoken/typed explanation. This is feedback, not replacement first evidence.
5. **22–25:** Use the [public teacher key](teacher/KEY-AND-NEXT.md) to select one next move. **Move:** if a prompt was known, recheck later with fresh locally chosen numbers.

**Alternative domain:** Fictional sports-facility adult role after the check. **Optional/home:** explain why pay for one shift and one month cannot be compared without the time basis.

## Day 16 · A budget keeps essentials visible

**Goal:** classify Card D's fixed, variable-essential and discretionary costs before finding an unassigned model balance. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Say Card A's $1,260 gross weekly figure is **stipulated as available in this exercise only**; a real budget must verify deductions. No private family budget is requested.
2. **2–7:** Model fixed `520+120+40=$680`; food is variable **essential**, not discretionary. Model `680+220+65+45=$1,010` planned.
3. **7–13:** Learners calculate discretionary `65+45=$110`, unassigned `1,260−1,010=$250` and reverse check `1,010+250=1,260`.
4. **13–20:** **Routes:** move labelled cost counters on the [budget mat](print/budget-map.pdf); fill subtotal boxes in large print; enter or dictate `income − fixed − variable essential − discretionary`. All routes preserve food's category and say what a real budget still needs.
5. **20–25:** Exit “Is $250 proven available in a real household?” **Key:** no; gross/net and unlisted costs remain unknown. **Move:** if food is called discretionary, ask whether the classification matches the stated essential role before recomputing.

**Alternative domain:** Imaginary community workshop operating plan. **Optional/home:** sort a made-up $900 income and three invented costs; no personal money disclosure.

## Day 17 · The GST base changes with the price label

**Goal:** calculate included and excluded GST in explicitly fully taxable fictional supply examples. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read Card E's condition **fully taxable** and the [ATO-source note](SOURCE-AND-CROSSWALK.md). The 10% rate is verified for a taxable supply; a real item's treatment needs its own check.
2. **2–7:** Model $84 **excluding** GST: `$84×0.10=$8.40`; inclusive `$92.40`. Check `84×1.10=92.40`.
3. **7–13:** For $121 **including** GST, model reversal `121÷1.10=$110` exclusive, GST `$11`; check `110+11=121`. Explain why `$121−10%×121` uses the wrong base.
4. **13–20:** **Routes:** move base/tax/total blocks with labelled denominators; complete the [price-base ladder](print/price-base.pdf); write/spoken calculation with factor 1.10 and reverse addition. Each route states the fully-taxable assumption and no real classification claim.
5. **20–25:** Exit “Is 10% of $121 the included GST?” **Key:** no; for this fully taxable example it is $11, one eleventh of inclusive price. **Move:** reconstruct base×1.10=total with $110 before a shortcut.

**Alternative domain:** Fictional adult poster-print supply. **Optional/home:** under an explicitly fully taxable **invented** example, find inclusive $55 from exclusive $50; no shopping task.

## Day 18 · Simple interest is not compounding

**Goal:** apply `I=Pin` with a yearly rate and labelled time, then test a changed term. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Name P, annual i and years n in Card F. Say these are fictional assumptions, not a real account or return.
2. **2–7:** Model `I=2400×0.035×2=$168`; final amount `$2,568`. Check 3.5% of 2400 is $84 for one year, so two simple-interest years give $168.
3. **7–13:** Learners recompute one year: interest $84/final $2,484. Explain why multiplying the first year's new amount by 3.5% would be **compounding**, which is not this stated model.
4. **13–20:** **Routes:** two equal $84 annual strips attached to original P only; fill the [interest timeline](print/interest-timeline.pdf); use symbolic `P×i×n` then verbalise units and no-compounding limit. All routes compare one and two years.
5. **20–25:** Exit “Which amount is the base in year 2 of this simple model?” **Key:** original $2,400. **Move:** if learner compounds, draw both $84 strips from the same P.

**Alternative domain:** Fictional invoice late-charge maths without a real contract. **Optional/home:** find `I` for invented P=$1,000, i=2%, n=2 years; no financial recommendation.

## Day 19 · Repeated computations must keep the same formula

**Goal:** audit three Card G budget what-ifs with one visible formula and check a numerical constraint. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Ask which inputs change by fictional week and which formula stays fixed. No one provides personal data.
2. **2–7:** Model Week A `1260−680−220−110=$250` unassigned; reverse check `680+220+110+250=1260`.
3. **7–13:** Learners calculate Week B `$205` and Week C `$245`, then check each against the invented **at least $200** constraint. Show where a spreadsheet could repeat `=income-fixed-essential-discretionary`; a paper table is equivalent for arithmetic.
4. **13–20:** **Routes:** budget counters and a $200 boundary strip; [budget map](print/budget-map.pdf) for each week; optional [local what-if spreadsheet](print/budget-what-if.xlsx) or typed/paper table with the same formula and a manual spot check. Every route reports all three values, one reverse check and an assumption limit. Actual spreadsheet use is documented separately.
5. **20–25:** Exit “If Week B food rises $10 alone, what happens?” **Key:** unassigned falls from $205 to $195, failing the stated model threshold. **Move:** if a student edits multiple cells, freeze unchanged inputs and subtract the one new $10.

**Alternative domain:** Invented stage-prop materials plan. **Optional/home:** change just one made-up cost by $5 and predict the balance; no family budget.

## Day 20 · Fresh public Check B

**Goal:** transfer GST-base and simple-interest reasoning to [new independent values](STUDENT-CHECKS.md#check-b--day-20--fully-taxable-print-and-simple-interest). **25 = 2 + 3 + 12 + 5 + 3.**

1. **0–2:** Explain the new case is public practice and its tax and interest assumptions are stipulated; it is not a QCAA exam or real advice.
2. **2–5:** Offer exact reading, large text, calculator, tactile factor cards, AAC/sign or exact-word scribe without giving the method for these new values.
3. **5–17:** Learners show first independent reasoning; collect before content feedback.
4. **17–22:** After collection, revisit `exclusive or inclusive base? / annual rate and years? / model or reality?` **Routes:** price blocks; annotated equation table; spoken/typed two-case audit. Preserve first evidence and record any hint.
5. **22–25:** Use the [public worked key](teacher/KEY-AND-NEXT.md) to choose a specific next practice. **Move:** create a new local parallel check when prior access made the first response ambiguous.

**Alternative domain:** Fictional sports-poster supply and a completely invented interest table after assessment. **Optional/home:** explain the difference between a percentage on a base and an included total.