# Year 8 number and sampling cards

All cards are constructed learning data. Use paper, counters, spreadsheet without an account, a tactile graph or plain text. Keep the qualifier **fictional** when a table is shown away from its lesson.

## Week 1: repair counts and model totals

`n` is the number of repairs, restricted to non-negative whole numbers. Plan A: `A(n)=18+7n`. Plan B: `B(n)=10n`. Amounts are invented Australian-dollar teaching values and exclude all other possible costs.

| n repairs | A: 18+7n | B: 10n | Lower model total |
| ---: | ---: | ---: | --- |
| 0 | $18 | $0 | B |
| 2 | $32 | $20 | B |
| 4 | $46 | $40 | B |
| 6 | $60 | $60 | equal |
| 8 | $74 | $80 | A |

Difference `A(n)-B(n)=18-3n`. Equal when `18-3n=0`, so `n=6`, verified by `18+7×6=60=10×6`. For whole `n<6` B is lower; for whole `n>6` A is lower. If a student draws continuous lines to see the algebra, mark **only integer points as possible repair counts**. Joining the dots helps show trend but half a repair is outside this model.

**Fresh Day 5 check:** imaginary art studio option C has $12 entry plus $5 per visit, `C(v)=12+5v`; D costs $8 per visit, `D(v)=8v`. C and D meet at `v=4`, both $32. At 3 visits C=$27, D=$24; at 6 visits C=$42, D=$48. `v` is a non-negative whole number; real service conditions are not supplied.

## Week 2: invented travel population and samples

Categories are neutral labels, not a hierarchy: `walk/roll | bus | bicycle | other`. “Walk/roll” can include mobility aids; do not infer individual ability. Imagined population of 100: `40 | 30 | 20 | 10`. Simulated simple random sample A of 20: `8 | 7 | 3 | 2`; simulated simple random sample B of 20: `10 | 5 | 4 | 1`. Constructed convenience sample C, 20 beside bicycle racks: `1 | 1 | 16 | 2`. All people and counts are fictional; the two random simulations are reproducible below and are not real surveys.

| Source | Walk/roll | Bus | Bicycle | Other | Total | Bicycle share |
| --- | ---: | ---: | ---: | ---: | ---: | ---: |
| Model population | 40 | 30 | 20 | 10 | 100 | 20% |
| Random model A | 8 | 7 | 3 | 2 | 20 | 15% |
| Random model B | 10 | 5 | 4 | 1 | 20 | 20% |
| Rack convenience C | 1 | 1 | 16 | 2 | 20 | 80% |

Two same-size random model samples need not have identical proportions; here A and B differ by five percentage points for bicycle. The rack sample is selected from a location linked to cycling, so it should not be used to estimate the whole school. The constructed population is known only so students can inspect error; in real investigations it is usually unknown.

**Optional sample-size simulation for Day 9:** Using the same fictional 100 IDs and the selection method below, run seeds `0–999` for sample sizes 10 and 40. For each simulated sample, calculate how many percentage points its bicycle share is from the known model 20%, then average those 1000 distances. The averages are **8.95 percentage points** for size 10 and **3.92 percentage points** for size 40. This particular repeated simulation illustrates that larger fair samples often vary less; it does **not** mean every size-40 draw is closer than every size-10 draw, or that a real population is known. Learners can compare this reported summary without programming.

**Fresh Day 10 check, different domain:** imaginary school arts club has a model population of 100 students, 60 interested in a new music room session and 40 not interested. Two seeded pseudorandom simulations of 20 report `11 interested/9 not` and `14 interested/6 not` (55% and 70%). A constructed convenience sample of 20 taken at a music-club meeting reports `18 interested/2 not` (90%). The samples differ; the meeting sample is selected where interested people are likely overrepresented. No real student interests are being recorded.

## How the simulated random samples were made

Assign fictional travel IDs `0–39` to walk/roll, `40–69` bus, `70–89` bicycle and `90–99` other. Python's `random.Random(seed).sample(range(100), 20)` selects 20 distinct model IDs. Seed `1` gives travel A's counts `8, 7, 3, 2`; seed `67` gives travel B's `10, 5, 4, 1`. For the fictional arts population, IDs `0–59` mean interested and `60–99` not interested. Seeds `5` and `4` give `11/9` and `14/6`. These are reproducible **pseudo**random simulations, not claims about students. The rack/meeting convenience examples are deliberately constructed contrasts, not outcomes of this mechanism.

**Rights:** Original fictional values and cards © NeuroForgeIO Pty Ltd 2026, SubjectNest, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/); credit, source and changes required.