Queensland Essential… | Year 11 Essential Maths · T1 W1–2 · Lessons

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 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 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, selected A4 and exact text/tactile aids, plain tokens or paper sketches, calculator where useful, 20 optional context swaps, and optional offline ratio lab 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.

0–2: Show three round and two square paper pieces. Ask, “Which group does the first 3 count?”
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.
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.
13–20: Routes: move labelled round/square tokens on the order mat; 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.
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.

Day 2 · Reversing a ratio changes its sentence

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

0–2: Read four amber and six blue cue cards aloud; offer texture labels so colour is not required.
2–7: Model amber:blue 4:6 and blue:amber 6:4. Write “for every” sentences and show the physical tray has not changed.
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.”
13–20: Routes: switch between the two labelled panels of the ratio order mat; 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.
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.

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.

0–2: Show eight story and twelve information badge tokens; ask for the total before any fraction.
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.
7–13: Learners simplify fractions to 2/5 and 3/5, check they sum to one, then note story:information simplifies to 2:3.
13–20: Routes: place eight and twelve tokens into the part-whole tray; 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.
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.

Day 4 · Simplifying keeps the relationship

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

0–2: Show 4:6 beside 2:3; ask what operation could link both numbers.
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.
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.
13–20: Routes: regroup counters into equal 2-or-4-part bundles on the simplify ladder; complete paired divide/multiply arrows; use offline ratio lab or type/speak two equivalent ratios and their common factor. All routes prove both terms changed by the same factor.
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.

Day 5 · Fresh public Check A

Goal: independently transfer order, simplest form and part-of-all reasoning to new marker data. 25 = 2 + 3 + 12 + 5 + 3.

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.
2–5: Give the exact prompt with usual neutral access supports, but no demonstration with its new values.
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.
17–22: After collection, compare a different two-shape example, explicitly naming the total before a fraction.
22–25: Choose one next move from the public teacher key. 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.

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.

0–2: Read 150 cm : 1 m. Ask what could go wrong if 150 and 1 are simplified directly.
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.
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.
13–20: Routes: line up labelled length strips on the unit mat; 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.
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.

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.

0–2: Read “large:small = 2:3” and total 35. Ask how many equal ratio parts there are altogether.
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.
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.
13–20: Routes: distribute 35 counters into five equal labelled trays using the split bar; fill a 5-part strip then write counts; use offline ratio lab or voice/type/AAC equal-part and total equations. Every route provides both reverse checks.
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.

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.

0–2: Read “1 cm paper : 200 cm model wall”; name drawing and wall sides. It is an invented sketch, not a real measurement.
2–7: Model 1:200 in cm; for 4 cm on paper, 4×200=800 cm=8 m on model wall. Reverse 800÷200=4 cm.
7–13: Learners calculate 3 cm plan →600 cm=6 m; explain why the 200 applies to every paper centimetre.
13–20: Routes: extend the scale ribbon 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.
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.

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.

0–2: Put Card G's three statements on the board without answers. Ask what evidence would test each.
2–7: Model 12:18 ÷6=2:3 and total30; show green share 12/30=2/5, not 2/3.
7–13: Learners rescale to total40: five parts, each8, giving16 green and24 orange; reverse-check16+24=40 and16:24=2:3.
13–20: Routes: use two textures of tag counters and five equal trays on the split bar; 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.
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.

Day 10 · Fresh public Check B

Goal: independently transfer fixed-total ratio division and simple scale reasoning to new fictional pack and plan data. 25 = 2 + 3 + 15 + 3 + 2.

0–2: Say this is a public formative check, not a secure exam, real plan or school/QCAA assessment instrument. Save first responses.
2–5: Give exact prompt, calculator and neutral access supports without solving its new ratio or scale.
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.
20–23: After collection, use different invented values to revisit total parts and cm-to-m conversion.
23–25: Use the public staff key 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.