# Mathematics checks: keys, misconceptions and rechecks

Keep this as a **formative teaching record**, not a grade or ability label. Everyone attempts the exit or independent check. Observe 2–3 planned focus learners per day and rotate; a partner result does not show independent reasoning. Record `independent / prompted / not yet observed`, exact work, access route and next move. Do not publish identifiable learner work.

| Day | Keyed exit | If the answer is incomplete, try this next |
|---:|---|---|
| 1 | Table B pairs distance (km) and duration (min); in this fictional set duration generally rises with distance. | If “distance causes exactly 35 minutes”, ask what traffic/stops and real observations are missing. |
| 2 | Trial 5 is `(3 N, 5.9 m/s²)`; force is horizontal. | If coordinates reverse, trace the axis labels and read trial 5 from the table again. |
| 3 | Strong positive, approximately linear association in **simulated** Table A; no finish comparison. | If “perfect” or “finish caused it”, point to repeated y-values and absent finish variable. |
| 4 | 8.1 is at 4 N, 1.8 at 1 N; ratio 4.5 compares changed forces, so does not isolate finish. | If student accepts headline, ask “Where are same-force different-finish trials?” |
| 5 | Check A below. | Separate plot-reading, association and claim-scope feedback. |
| 6 | `B(m)=480+50m`; 480 is setup dollars, 50 dollars per complete month; B(4)=$680. | If 480 is multiplied by 4, ask what is paid once. |
| 7 | Crossing `(8 months,$880)`; both equations substitute to $880. | If 8 has no units, interpret both coordinates and note equality is only price. |
| 8 | A: `m≤9.5`, max 9 whole months; A(9)=$960, A(10)=$1,040. B max 10. | If 9.5 is given as a payment count, reread whole-month domain and boundary values. |
| 9 | `B_new=420+50m`, crossing at 6 months and $720; changed setup is $60 lower. | If crossing stays 8, substitute 8 into modified B and compare. |
| 10 | Check B below. | Decide whether error is equation, algebra, cap interpretation or missing assumption. |

## Check A — Day 5 statistics and media claim

**Prompt:** On axes with distance 0–12 km and duration 0–40 minutes, plot or accurately describe Table B's `(2,7),(4,13),(8,26),(12,35)`. Describe direction and approximate form. Repair: “Every real 12 km trip takes exactly 35 minutes.”

**Key:** Points at the four stated coordinates; **positive, roughly linear** pattern in the fictional pairs. A defensible repair is “In this invented route list, the 12 km route is listed as 35 minutes; the data cannot predict all real trips.” An accessible text description naming paired values and association is equivalent evidence to plotting when the visual/motor step is not the target. If a point is `(7,2)`, the axes likely reversed. If the answer is “perfect straight line,” ask the student to compare slopes between adjacent points. If scope is missing, ask who/what was actually measured: nobody; the dataset is authored.

**Independent recheck (later, not in the same 3-minute feedback window):** New invented pairs `(1,4),(3,10),(5,16),(7,22)` for quantity x and time y. Describe association and repair “This proves every real process takes 22 units at x=7.” Key: positive **exactly linear within the four constructed points**, slope 3 units of y per unit x, but no general real-world proof. Use x/y units supplied in the local prompt or call them abstract quantities; do not invent a physical context.

## Check B — Day 10 linear model and constraint

**Prompt:** X has $180 setup plus $90 for each complete month; Y has $420 setup plus $60/month; `m` is whole months 0–12. Find model equations, equal-cost point, both 10-month totals, whether either fits a **$1,000** practice cap, and one assumption to verify.

**Key:** `X(m)=180+90m`; `Y(m)=420+60m`. Equality: `180+90m=420+60m` → `30m=240` → `m=8`; both cost **$900**. At 10 months, X **$1,080** and Y **$1,020**; **neither** fits $1,000. A legitimate missing issue is actual terms, device condition, exit fee, access/service quality or rights. “Y is cheaper at ten months” is true **within the model** but does not make Y affordable under the stated cap. Reject a real purchase recommendation from this sheet.

**Worked learner patterns and next move:**

- `m=8` with totals $900 for both: secure intersection, but still ask for interpretation “8 complete months, equal modelled cost”.
- `m=240/30` but writes 30 months: substitution reveals the error; try an unseen two-plan equality.
- Correct X/Y totals and “choose Y” without cap check: put 1,020 and 1,000 on a number line, then reframe the decision.
- Correct algebra using a calculator under an access plan: keep the equation/substitution evidence and tool note; do not treat calculation aid as conceptual failure.
- Says price proves quality: ask what the brief omits, and separate computed cost from unverified service attributes.

**Independent algebra recheck:** Two new fictional plans `R(m)=200+70m`, `S(m)=350+50m`, whole months 0–12. Equality gives `20m=150` → **m=7.5**, cost **$725**. Because the model allows only whole months, **no complete-month point has exactly equal cost**; at month 7 R=$690 and S=$700, while at month 8 R=$760 and S=$750. This probes whether the learner can interpret a continuous graph against a discrete domain. Use only after a relevant teaching move, not as an immediate duplicate of Day 10.

## Local record strip

`date | learner code (local only) | task | exact evidence | independent/prompted/not yet observed | support/tool | next move | recheck evidence`. Keep under school policy. No child data is sent to SubjectNest.

**Reuse:** Original assessment and fictional data © NeuroForgeIO Pty Ltd 2026, SubjectNest, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). Credit source/licence and mark changes.