# Staff keys, criteria and next teaching

Keep this file separate from learner first sources and reserved rechecks. A learner who has seen a source or its answers can still learn from it; their later response cannot be described as fresh. Use an unused recheck after specific teaching if a further fresh response is useful.

## E · Full solved key

**Part 1.** There are 3 + 2 = 5 equal parts. Each part represents 50/5 = 10 symbols. There must be 30 stars and 20 circles. Star fraction of all symbols is 30/50 = 3/5. Check both the total 30 + 20 = 50 and star:circle ratio 30:20 = 3:2.

**Part 2.** A unit price is $4/10 = $0.40/star. To obtain at least 30 stars, the cheapest sufficient A order is 3 whole packs: 30 stars, $12, no spare stars. B unit price is $2.10/6 = $0.35/star. The cheapest sufficient B order is 5 whole packs: 30 stars, $10.50, no spare stars. B meets the $11 budget; A exceeds it by $1. The initial recommendation is B, supported by the actual total order, not merely a claim that its unit price is lower. Useful assumptions include every star being usable, packs being whole, no first-stage fees and circles already being provided without cost. Extra packs cannot reduce either supplier's cost because every pack has a positive stated price.

**Part 3.** The new B total is $10.50 + $2 = $12.50. A remains $12. With the new $12.25 budget, A fits with $0.25 spare; B exceeds the budget by $0.25. Revise the recommendation to A. The new fee and new budget change the bounded decision. The star and circle counts do not change. A valid review states the changed condition, both revised totals and a conclusion supported by the new limit. Keeping the old recommendation with no recalculation would not fit this source.

## F · Full solved key

**Part 1.** Five equal parts share 30 segments, so each part contains 6 segments. Counts are 12 short and 18 long. Short fraction of the whole is 12/30 = 2/5. Short tape length is 12 × 3/4 = 9 m; long tape length is 18 × 5/6 = 15 m. Total is 9 + 15 = 24 m. An efficient equivalent calculation is six groups, each containing 2 × 3/4 + 3 × 5/6 = 3/2 + 5/2 = 4 m; six groups use 24 m. The count ratio alone does not give a 2:3 ratio of lengths, because segment lengths differ.

**Part 2.** A rate is $1.50/m and B $1.25/m. The needed 24 m is already a whole number of metre units, so each cheapest sufficient order contains 24 units with no spare length. A costs 24 × $1.50 = $36; B costs 24 × $1.25 = $30. B meets the $31 budget with $1 spare; A exceeds it by $5. Initial recommendation is B. The model assumes all tape is usable, joins are allowed, cutting loss is zero, quantities are sold in whole metre units and there are no first-stage fees. A learner who questions the usefulness of zero cutting loss has noticed a real model limit; they should still solve the stated model before recommending a further check.

**Part 3.** The new B total is $30 + $2 = $32. A remains $36. Under the new $32 budget, B fits exactly and A exceeds it by $4. Keep B after recalculation. The quantity 24 m and the segment counts remain unchanged. The fee is added once per order, not per metre. A valid KEEP decision identifies the changed fee and budget, gives both new totals and supports the retained recommendation. A guessed switch to A does not fit these conditions. If joins or cutting loss changed instead, the physical model would need more information; that is not the change stated here.

## Descriptive criteria

Use these descriptions to give next-step feedback. They are local formative criteria, not QCAA standard elaborations or an A–E conversion. Record the particular evidence and support rather than adding an invented overall score.

| Criterion | Clear evidence | Developing evidence | A useful next teaching step |
|---|---|---|---|
| Ratio and whole, E/F | Names the compared parts, uses the sum of ratio terms for the whole, finds exact counts and checks both sum and ratio | Correct counts with an unclear whole or fraction; or a sound diagram with a scaling error | Point to one complete group; label each part and the whole before scaling |
| Exact calculation, E/F | Records operations that match the quantities, calculates accurately and distinguishes amount from count; F includes exact rational lengths and their sum | Method fits but arithmetic slips; or an answer without a checkable method | Use equal parts or exact fractions, then check by an inverse or a second representation |
| Units and rate, E/F | States $/star or $/m, uses the rate on the correct quantity and keeps an order fee separate | Bare decimal price; reversed units; or fee multiplied by every item/metre | Say “dollars for one ___”, label both units and put the one-order fee on its own line |
| Conditions and initial decision, E/F | Finds the cheapest sufficient whole-pack/whole-metre order for each supplier, compares actual totals with the budget and names a relevant assumption | Chooses from unit prices alone, ignores a whole-unit constraint or gives a budget claim with no totals | Build an order table: need, quantity per unit, whole units, total, spare, budget |
| Review and interpretation, E/F | Keeps the initial response, recalculates both totals under the new fee/budget, identifies what stayed the same and revises to a supported decision | Notes that something changed but does not update the model; or erases the first response | Circle changed source quantities, keep unchanged quantities, recompute before deciding |

For F, the exact-calculation description includes multiplying the number of each segment by its own length and adding lengths in the same unit. Multiplying a total count by just one segment length is an incomplete model even if its arithmetic is accurate. For E, the exact-calculation description concerns scaling, unit prices and total costs; do not infer mastery of later fraction operations from E.

## Four reteach branches

**R1 · Whole and ratio.** Return to Day 21A or 22B with different ordinary numbers. Ask the learner to enclose one complete group, count all parts and label part:part versus part:whole. Try a total that fits, then a total that cannot fit whole counts. After teaching, use unused recheck E1 if appropriate.

**R2 · Rate versus whole order.** Return to Day 24A. Make a table with need, pack size, unit price, packs, supplied quantity, total and spare. Keep any fee in a separate once-per-order field. Compare the budget with the order total. After teaching, use unused recheck E2 if appropriate.

**R3 · Exact rational quantities.** Return to Day 27A or 28C, then Day 29A. Name the whole and unit, convert mixed numbers, cancel factors when multiplying, and use a common denominator when adding. Check the model as well as the arithmetic. After teaching, use unused recheck F1 if appropriate.

**R4 · Review a condition.** Return to Day 24B or 29C. Keep the initial model, highlight only changed information and recompute each affected total. Ask why unchanged counts should stay unchanged. After teaching, use unused recheck F2 if appropriate.

For integer-sign or order-of-operation uncertainty, use Day 26B/C and W12–14. The E/F rechecks do not replace an integer-operation response. Do not claim that passing them proves the full AC9M8N04 repertoire.

## Collection and support record

Record: date; source identifier; whether source/key was unseen; time for first stage; first response kept; time of change-card reveal; time for review; tools; blank supports; spoken/scribed assistance; neutral prompts; learner's mathematical choices; next teaching step; unused recheck selected. A staff model is authored illustrative work. It is not a learner response, human trial, spoken performance or independently observed outcome.