# Queensland Year 11 Essential Mathematics · Unit 1 Topic 1 ratio starter

This is a **separate Applied (Essential) pathway**, not the General Mathematics course and not a claim that all Year 11 learners take it. The current official [QCAA Essential Mathematics 2025 v1.3 syllabus](https://www.qcaa.qld.edu.au/downloads/senior-qce/syllabuses/snr_ess_maths_25_app_syll.pdf) places **Unit 1: Number, data and money**, the Fundamental topic **Calculations**, and **Topic 1: Number** at PDF pp. 16–17 / printed pp. 14–15. Topic 1 starts with **Ratios (7 hours)**. These ten 25-minute sessions total **4 hours 10 minutes**, so they are a **partial ratio starter**, not the 7-hour subtopic, Unit 1, a school formal assessment or a whole Year 11 course. The [source-page crosswalk](SOURCE-AND-CROSSWALK.md) states precisely what is and is not covered. `EM-U1-T1-R` is SubjectNest shorthand only, **not a QCAA or national code**.

**Prepare:** [Cards A–G](LEARNER-CARDS.md), selected [A4 and exact text/tactile aids](print/TEXT-ALTERNATIVES.md), plain tokens or paper sketches, calculator where useful, [20 optional context swaps](PRACTICE-SWAPS.md), and optional [offline ratio lab](ratio-lab.html) with a paper equivalent. No money or private learner information is required. Every route shows the same mathematical target through objects, written table/diagram, or voice/AAC/typing. Record mathematical prompts separately from access supports. The ordinary planned clock is **2 launch + 5 explicit model + 6 guided + 7 independent choice + 5 exit/response = 25 minutes**. Day 5 uses **2 + 3 + 12 + 5 + 3 = 25**; the longer Day 10 transfer uses **2 + 3 + 15 + 3 + 2 = 25**. Actual classroom timing is untested.

## Day 1 · A ratio names two ordered parts

**Goal:** state Card A's round:square ratio with labels and compare each part with the whole. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Show three round and two square paper pieces. Ask, “Which group does the first 3 count?”
2. **2–7:** Model **round:square = 3:2**. Say “three round for every two square,” then count five altogether. Draw a boundary between part-to-part `3:2` and part-to-all `3:5`.
3. **7–13:** Learners physically or visually swap the order: **square:round = 2:3**, while the total stays five. Ask each to point to the named first group.
4. **13–20:** **Routes:** move labelled round/square tokens on the [order mat](print/ratio-order.pdf); draw two labelled rows and write both ratios; speak/type/AAC “first group : second group” and give the five-token check. All routes identify which comparison was made.
5. **20–25:** Exit “If I say square:round, why is 3:2 wrong?” **Key:** the first number must count squares, so 2:3. **Move:** underline the first named group before writing any numbers.

**Alternative domains:** invented poster icons or repair-kit cards. **Optional/home:** sketch a fictional 4 stars and 1 line, name both ratio orders; no collection from home. [Two more cases](PRACTICE-SWAPS.md#day-1--name-both-parts).

## Day 2 · Reversing a ratio changes its sentence

**Goal:** interpret Card B's two orderings before simplifying. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read four amber and six blue cue cards aloud; offer texture labels so colour is not required.
2. **2–7:** Model amber:blue `4:6` and blue:amber `6:4`. Write “for every” sentences and show the physical tray has not changed.
3. **7–13:** Learners decide whether `4:6` could answer a question explicitly asking blue:amber. They must cite the labels rather than only saying “no.”
4. **13–20:** **Routes:** switch between the two labelled panels of the [ratio order mat](print/ratio-order.pdf); annotate `amber / blue` then `blue / amber` table rows; speak, type or AAC two ordered ratio sentences. Each route includes a ten-card total check and says why order matters.
5. **20–25:** Exit “Does `6:4` mean more cards were added?” **Key:** no, it describes the same tray in reversed order. **Move:** compare the two named group labels before comparing numbers.

**Alternative domains:** stage marks or invented garden markers. **Optional/home:** write two orderings for fictional 2 circle and 5 triangle symbols. [Two more cases](PRACTICE-SWAPS.md#day-2--order-changes-meaning).

## Day 3 · A part-to-part ratio is not a part-to-whole fraction

**Goal:** use Card C to connect ratio and fractions without swapping denominators. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Show eight story and twelve information badge tokens; ask for the total before any fraction.
2. **2–7:** Model story:information `8:12`; story fraction of **all** `8/20`; information fraction `12/20`. The colon compares groups, and a fraction names a part of a specified whole.
3. **7–13:** Learners simplify fractions to `2/5` and `3/5`, check they sum to one, then note story:information simplifies to `2:3`.
4. **13–20:** **Routes:** place eight and twelve tokens into the [part-whole tray](print/part-whole.pdf); shade two sections of a 20-cell strip and label denominators; voice/type/AAC `8:12`, `8/20`, `12/20` with a sentence naming each whole. All routes verify 8+12=20.
5. **20–25:** Exit “Why isn't story's share `8/12`?” **Key:** 12 is the other part, not all 20 badges. **Move:** circle the phrase “of all” and count the total first.

**Alternative domains:** fictional map markers or studio cue cards. **Optional/home:** use invented 2 red and 3 plain shapes to show ratio and red fraction of all. [Two more cases](PRACTICE-SWAPS.md#day-3--part-to-part-versus-part-to-all).

## Day 4 · Simplifying keeps the relationship

**Goal:** find Card B's simplest whole-number ratio and verify equivalence. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Show `4:6` beside `2:3`; ask what operation could link both numbers.
2. **2–7:** Model dividing both parts by common factor 2: `4÷2:6÷2 = 2:3`. Reverse by multiplying each part by 2. Keep the ten actual cue cards visible.
3. **7–13:** Learners simplify Card C's `8:12` by 4 to `2:3` and explain why dividing only the first term is invalid. Use a calculator only as a check.
4. **13–20:** **Routes:** regroup counters into equal 2-or-4-part bundles on the [simplify ladder](print/simplify-ladder.pdf); complete paired divide/multiply arrows; use [offline ratio lab](ratio-lab.html) or type/speak two equivalent ratios and their common factor. All routes prove both terms changed by the same factor.
5. **20–25:** Exit “Does simplifying `4:6` to `2:3` discard cards?” **Key:** no, it expresses the same relation in lowest whole-number terms. **Move:** reconstruct `4:6` from `2:3` to show equivalence.

**Alternative domains:** imaginary library badges or game tiles. **Optional/home:** simplify a made-up `10:15` symbol mix; no real counts. [Two more cases](PRACTICE-SWAPS.md#day-4--simplify-without-removing-objects).

## Day 5 · Fresh public Check A

**Goal:** independently transfer order, simplest form and part-of-all reasoning to [new marker data](STUDENT-CHECKS.md#check-a--day-5--order-simplest-form-and-part-of-all). **25 = 2 + 3 + 12 + 5 + 3.**

1. **0–2:** Explain that this public check is fresh relative to cards but **not a secure exam or QCAA school assessment**. Collect first independent work before feedback.
2. **2–5:** Give the exact prompt with usual neutral access supports, but no demonstration with its new values.
3. **5–17:** **Routes:** handle labelled triangle/circle counters; annotate ratio/fraction boxes on blank paper; type, write, dictate or AAC the same labelled ratios, fractions and check. Preserve first work and note any content cue separately.
4. **17–22:** After collection, compare a different two-shape example, explicitly naming the total before a fraction.
5. **22–25:** Choose one next move from the [public teacher key](teacher/KEY-AND-NEXT.md). **Move:** if prior answer access matters, set a new local parallel prompt.

**Alternative domains:** invented gallery icons or stage symbols **after** collection. **Optional/home:** explain the difference between `A:B` and `A:(A+B)` with made-up counts. [After-check swaps](PRACTICE-SWAPS.md#day-5--after-check-only).

## Day 6 · Compare like lengths in like units

**Goal:** convert Card D's lengths to a shared unit before making a simplest ratio. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read `150 cm : 1 m`. Ask what could go wrong if 150 and 1 are simplified directly.
2. **2–7:** Model `1 m=100 cm`; now `150 cm:100 cm=150:100=3:2`. Check `3×50 cm=150 cm`, `2×50 cm=100 cm`.
3. **7–13:** Learners compare a second invented pair `60 cm:40 cm=3:2` and explain why the values differ but the ratio matches. The physical lengths remain unequal.
4. **13–20:** **Routes:** line up labelled length strips on the [unit mat](print/simplify-ladder.pdf); fill a convert-then-divide table; calculate and voice/type/AAC both-unit and simplified statements. All routes identify the common unit and verify by scaling back.
5. **20–25:** Exit “Is `150:1` the simplest length ratio here?” **Key:** no; units differ, and after conversion the ratio is 3:2. **Move:** rewrite both lengths in cm before any cancellation.

**Alternative domains:** fictional ribbon display or model path. **Optional/home:** compare invented 2 m and 1 m in centimetres, then simplify. [Two more cases](PRACTICE-SWAPS.md#day-6--match-units-first).

## Day 7 · A fixed total can be divided into ratio parts

**Goal:** allocate Card E's 35 identical labels in a 2:3 ratio with total and relationship checks. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read “large:small = 2:3” and total 35. Ask how many equal ratio parts there are altogether.
2. **2–7:** Model `2+3=5` parts; `35÷5=7` labels per part; large `2×7=14`, small `3×7=21`. Check `14+21=35` and `14:21=2:3`.
3. **7–13:** Learners find how a **second** fictional total 25 would split under the same ratio (10 and15) and explain why they cannot simply give 2 and3 labels for total35.
4. **13–20:** **Routes:** distribute 35 counters into five equal labelled trays using the [split bar](print/split-bar.pdf); fill a 5-part strip then write counts; use [offline ratio lab](ratio-lab.html) or voice/type/AAC equal-part and total equations. Every route provides both reverse checks.
5. **20–25:** Exit “Why is 35÷2 the wrong first division?” **Key:** total comprises 2+3=5 parts. **Move:** draw exactly five equal boxes before sharing any labels.

**Alternative domains:** made-up event signs or archive tags. **Optional/home:** split 20 imaginary tiles 1:3 and check. [Two more cases](PRACTICE-SWAPS.md#day-7--split-a-total-by-parts).

## Day 8 · A simple scale connects paper and model distance

**Goal:** use Card F's drawing:wall scale to convert plan lengths with matching units. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Read “1 cm paper : 200 cm model wall”; name drawing and wall sides. It is an invented sketch, not a real measurement.
2. **2–7:** Model `1:200` in cm; for 4 cm on paper, `4×200=800 cm=8 m` on model wall. Reverse `800÷200=4 cm`.
3. **7–13:** Learners calculate 3 cm plan →600 cm=6 m; explain why the 200 applies to every paper centimetre.
4. **13–20:** **Routes:** extend the [scale ribbon](print/scale-ribbon.pdf) four equal paper steps with 200-cm model blocks; annotate a two-column plan/wall table; use unit-labelled multiplication and speak/type/AAC the conversion and reverse. All routes state both units and the fictional-plan limit.
5. **20–25:** Exit “Does 4 cm on the paper mean 800 m?” **Key:** no; first 800 **centimetres**, then 8 metres. **Move:** attach the unit to the 200 before multiplication and divide by 100 only after.

**Alternative domains:** imaginary mural sketch or stage-floor model. **Optional/home:** draw 2 cm representing 100 cm per paper cm; label both units. [Two more cases](PRACTICE-SWAPS.md#day-8--a-drawing-scale).

## Day 9 · Audit a plausible ratio claim

**Goal:** use Card G to check equivalence, part-of-all and a rescaled fixed total, correcting unsupported claims. **25 = 2 + 5 + 6 + 7 + 5.**

1. **0–2:** Put Card G's three statements on the board without answers. Ask what evidence would test each.
2. **2–7:** Model 12:18 ÷6=2:3 and total30; show green share `12/30=2/5`, not `2/3`.
3. **7–13:** Learners rescale to total40: five parts, each8, giving16 green and24 orange; reverse-check16+24=40 and16:24=2:3.
4. **13–20:** **Routes:** use two textures of tag counters and five equal trays on the [split bar](print/split-bar.pdf); mark true/false plus corrected equations in an audit table; speak/type/AAC each claim, evidence and correction. All routes distinguish 2 of 5 total parts from 2 of 3 other-part comparison.
5. **20–25:** Exit “What denominator belongs to ‘green of all’?” **Key:** five ratio parts, or 30 actual tags here. **Move:** draw a brace around both groups before forming a part-of-all fraction.

**Alternative domains:** fictional library categories or festival wayfinding icons. **Optional/home:** audit an imagined claim that 2:3 with total15 means 2 and3 objects; show the actual counts. [Two more cases](PRACTICE-SWAPS.md#day-9--check-a-claim).

## Day 10 · Fresh public Check B

**Goal:** independently transfer fixed-total ratio division and simple scale reasoning to [new fictional pack and plan data](STUDENT-CHECKS.md#check-b--day-10--fixed-total-and-drawing-scale). **25 = 2 + 3 + 15 + 3 + 2.**

1. **0–2:** Say this is a **public formative check**, not a secure exam, real plan or school/QCAA assessment instrument. Save first responses.
2. **2–5:** Give exact prompt, calculator and neutral access supports without solving its new ratio or scale.
3. **5–20:** **Routes:** distribute counters into labelled equal-part trays and use a physical scale strip; complete blank ratio/plan tables; write, type, dictate or AAC the same equations, units and checks. Record any mathematical hint separately.
4. **20–23:** After collection, use different invented values to revisit total parts and cm-to-m conversion.
5. **23–25:** Use the [public staff key](teacher/KEY-AND-NEXT.md) for one targeted response move. **Move:** set a new local case if public-key access affects independence.

**Alternative domains:** imaginary model-garden signs or paper exhibit labels **after** collection. **Optional/home:** explain why a drawing ratio cannot verify a real wall's dimensions. [After-check swaps](PRACTICE-SWAPS.md#day-10--after-check-only).