# Public teacher-targeted worked key · Essential Mathematics ratio starter

This key is **publicly accessible by URL**. The two [learner checks](../STUDENT-CHECKS.md) are formative practice, **not secure exams, grades or QCAA/school Unit 1 assessment instruments**. Keep a first independent response before feedback, record content hints separately from access supports, and choose fresh local values if prior access matters. The cases are all invented; do not collect a learner's household, pay, disability or community data. Offer objects, large print, oral/AAC and written expression at the same target level.

## Daily mathematics and next response

| Day | Worked target | If a learner needs another step |
| --- | --- | --- |
| 1 | Card A round:square `3:2`, square:round `2:3`; total 5, round part of all `3/5`, square `2/5`. | Label first and second parts on tokens before writing the colon. |
| 2 | Card B amber:blue `4:6`, blue:amber `6:4`; ten cards in either description. | Keep the tray fixed and swap only the spoken/written order. |
| 3 | Card C story:information `8:12=2:3`; total20; story fraction `8/20=2/5`; information `12/20=3/5`. | Put a bracket around **all 20** before choosing a fraction denominator. |
| 4 | Card B `4:6÷2=2:3`; reverse `2×2:3×2=4:6`; Card C `8:12÷4=2:3`. Actual objects remain. | Apply the same nonzero factor to both parts and reverse to check. |
| 5 | See Check A below; keep first work before feedback. | Reteach the weak order, common-factor or part-of-all link with unseen numbers. |
| 6 | Card D `150 cm:1 m=150 cm:100 cm=3:2`; `3×50:2×50=150:100` reverse. | Write both values in cm before simplifying. |
| 7 | Card E total parts `2+3=5`; one part `35÷5=7`; large `14`, small `21`; checks `14+21=35`, `14:21=2:3`. For invented total25, 10 and15. | Draw five equal boxes, then place the total across them. |
| 8 | Card F paper:wall `1 cm:200 cm =1:200`; 4 cm→800 cm→8 m, 3 cm→600 cm→6 m. Reverse 800÷200=4. | Keep cm until the last conversion; ask which side of the colon is the drawing. |
| 9 | Card G statement 1 true: `12:18=2:3`; statement 2 false: green fraction `12/30=2/5`; statement 3 true: at total40, `16:24=2:3`. | Name the **other part** for a ratio and **all parts** for a fraction before checking. |
| 10 | See Check B below; do not treat a paper scale as a measured wall. | Recheck total parts or cm→m according to first-work evidence. |

## Check A · independent worked response

1. Triangle:circle `9:15`, divide both by 3 to get `3:5`. Circle:triangle `15:9`, divide by 3 to get `5:3`. The first number changes because the named first group changes; no markers change.
2. Total `9+15=24`. Triangle share `9/24=3/8`; circle share `15/24=5/8`. Check `3/8+5/8=1` and `3+5=8` simplified total parts.
3. Triangle:circle compares triangle with the other group; triangle:all compares triangle with both groups and is `9:24=3:8` or fraction `3/8`. Accept a clear total/addition check and word labels.

**Next moves:** If `9/15` is called triangle fraction of all, circle 24 as the total. If only one term is divided in the simplification, multiply back and compare the original. If ratio order reverses inadvertently, have the learner underline the first noun. Do not replace the first response after coaching.

## Check B · independent worked response

1. Plain:striped `4:7` has `4+7=11` parts. `55÷11=5` badges per part. Plain `4×5=20`, striped `7×5=35`. Check `20+35=55` and `20:35=4:7` after dividing both by 5.
2. Both units in cm: plan:wall `1:250`. A 6 cm plan line represents `6×250=1500 cm=15 m` on the invented model wall. Reverse `1500÷250=6 cm`.
3. `10 m=1000 cm`; plan length `1000÷250=4 cm`. Reverse `4×250=1000 cm=10 m`.
4. Accept: the badge counts and wall scale are invented; no actual stock, measurement accuracy, building constraints or safety has been checked.

**Next moves:** If a learner divides 55 by 4, draw 11 boxes. If 1500 is labelled metres, write the unit `cm` after 250 before multiplying, then convert. If 10 m is used directly with 250 cm, first change 10 m to 1000 cm. Record which step was cued and retest with a different local example.

## Twenty optional swap results

Two results per day correspond, in order, to the public [practice swaps](../PRACTICE-SWAPS.md). They are optional transfer cases, not additional formal assessment.

| Day | Swap 1 | Swap 2 |
| --- | --- | --- |
| 1 | Poster: triangle:circle `5:3`, reverse `3:5`; total8; fractions `5/8` and `3/8`. | Music: beat:rest `7:2`, total9. |
| 2 | Garden: tall:short `2:5`, reverse `5:2`. | Exhibit: square:round `3:8`, reverse `8:3`. |
| 3 | Map: bridge:path `2:3`; bridge:all `2:5`, fraction `2/5`. | Stage: star:line `4:1`; star fraction of all `4/5`. |
| 4 | Badge: `6:9=2:3` by dividing by 3. | Tool tags: `14:21=2:3` by dividing by 7. |
| 5 | Archive: `12:16=3:4`; blue fraction `12/28=3/7`. | Gallery: `5:15=1:3`; arrow fraction `5/20=1/4`. |
| 6 | Ribbon `120:80=3:2`. | Model path `2 m:50 cm=200 cm:50 cm=4:1`. |
| 7 | Kit: 8 parts, 5 each, counts `15` and `25`; sum40. | Stage: 3 parts, 12 each, counts `12` and `24`; sum36. |
| 8 | Sketch: 2 cm→600 cm=`6 m`. | Diagram: 6 cm→300 cm=`3 m`. |
| 9 | Tag audit: `2:3`, total50, one part10, counts `20` and `30`, not 25/25. | Reverse plan: 12 m=1200 cm, `1200÷200=6 cm` drawing. |
| 10 | Label split: 7 parts, 6 each, counts `12` and `30`; sum42. | Plan: 3 cm→1200 cm=`12 m`. |

## Criteria to observe, not a grade

Record whether the learner names the two groups in order, distinguishes other-part versus whole, converts to matching units, scales both ratio parts by the same factor, checks a split by both total and ratio, and labels drawing versus model units. Accept tactile, visual, written, verbal/AAC and calculator-assisted explanations that show the same reasoning. These short checks do **not** assess the whole 7-hour Ratios subtopic, rates/percentages/energy, other Unit 1 topics or school formal assessment objectives. No real-class learning benefit has been observed.