# Ten Year 8 mathematics lessons, 25 minutes each

Use [fictional number cards](MATERIALS.md), [graph](../print/plan-graph.svg), [sample grid](../print/sample-grid.svg) and [keys](ASSESSMENT.md). Every lesson has **2 + 4 + 5 + 7 + 4 + 3 = 25 minutes**. Maths may be shown with an equation, table, tactile tokens, graph, calculator/spreadsheet or spoken argument; record the route. Selected partial codes: `AC9M8A01`, `AC9M8A02`, `AC9M8A03`, `AC9M8ST01`, `AC9M8ST02`, `AC9M8ST03`, `AC9M8ST04`.

## Day 1 — Name the variable before calculating

**Success:** express a fixed-plus-variable cost and state a sensible domain. (`AC9M8A01`, `AC9M8A03`)

1. **Read · 2 min.** Show Plan A's $18 joining cost and $7 per repair. Ask which cost changes when repairs change.
2. **Model · 4 min.** Let `n` be repairs. Write `A(n)=18+7n`. The 18 is fixed, the 7 multiplies the number of repairs; `n=0,1,2…`, not negative or half a repair.
3. **Guided build · 5 min.** Learners form `B(n)=10n` from Plan B's no-joining-cost card. Substitute `n=2`: A=$32, B=$20.
4. **Independent turn · 7 min.** For `n=3`, learners compute A=$39, B=$30 and write what each term means. Paper, tokens, speech or keyboard accepted.
5. **Error check · 4 min.** Repair `18n+7` by testing zero repairs: the incorrect expression yields $7, whereas the joining fee alone would be $18.
6. **Exit · 3 min.** “What does 7 represent in A?” Key: $7 for each repair, not $7 fixed joining cost.

## Day 2 — A table is a model, not a quotation

**Success:** calculate a table and identify what the model omits. (`AC9M8A03`)

1. **Recall · 2 min.** Ask for A and B at zero repairs: $18 and $0.
2. **Model · 4 min.** Fill `n=2` and `n=4`: A $32/$46, B $20/$40; use substitution so each entry can be checked.
3. **Guided finish · 5 min.** Class completes `n=6`: both $60. Ask why equality at one count does not make all counts equal.
4. **Independent table · 7 min.** Learners fill `n=8`: A $74, B $80, then compare at `n=0,2,4,6,8` in a sentence.
5. **Assumption scan · 4 min.** Circle omitted parts/quality/travel/eligibility. A correct model answer is not a real cheapest purchase claim.
6. **Exit · 3 min.** “At eight repairs, which model total is lower and by how much?” Key: A by $6.

## Day 3 — Plot the possible counts

**Success:** connect a table to a coordinate graph while respecting whole-number domain. (`AC9M8A02`, `AC9M8A03`)

1. **Orient · 2 min.** On [print axes](../print/plan-graph.svg), x is number of repairs and y is total dollars; ask what `(2,32)` means.
2. **Model · 4 min.** Plot A's `(0,18)`, `(2,32)`, `(4,46)` and B's `(0,0)`, `(2,20)`, `(4,40)`.
3. **Guided extend · 5 min.** Plot both at `n=6` and `n=8`. Locate meeting `(6,60)`.
4. **Independent statement · 7 min.** Learners write one sentence about slope per repair and one about intercept. A increases $7 per repair from $18; B increases $10 from $0.
5. **Domain check · 4 min.** If lines are drawn to show trend, circle integer x-values as actual cases. Ask why `n=4.5` is not a possible repair count here.
6. **Exit · 3 min.** “What does `(6,60)` mean?” Key: six repairs cost $60 under either supplied model.

## Day 4 — Solve, then substitute

**Success:** solve the equality and verify it in both formulas. (`AC9M8A01`, `AC9M8A02`)

1. **Question · 2 min.** Ask which `n` makes totals equal, without pointing to graph first.
2. **Model · 4 min.** Write `18+7n=10n`, subtract `7n` from both sides to get `18=3n`.
3. **Guided finish · 5 min.** Divide by 3: `n=6`. Substitute: A `18+42=60`, B `60`.
4. **Independent compare · 7 min.** Learners choose `n=5` and `n=7` to show direction changes: at 5, A=$53/B=$50; at 7, A=$67/B=$70.
5. **Method compare · 4 min.** Table, graph and algebra agree on six. Ask which one best exposes the exact equality and why.
6. **Exit · 3 min.** “Is A lower at five?” Key: no, A=$53, B=$50.

## Day 5 — Fresh linear check A

**Success:** model and compare a new fixed-plus-variable pair independently. Prepare [art-studio card](ASSESSMENT.md#day-5-check-a). (`AC9M8A01`, `AC9M8A02`, `AC9M8A03`)

1. **Set · 2 min.** State that fictional visits are whole, non-negative numbers; no real fee is being quoted.
2. **Clarify · 4 min.** Read only the new conditions: C $12 plus $5 per visit; D $8 per visit. Do not model the equation.
3. **Plan · 5 min.** Learners choose table, graph or equation, with enough room to show work.
4. **Independent response · 7 min.** Find formulas, equality point and which plan is lower at three and six visits.
5. **Self-check · 4 min.** Substitute the meeting value into both; check a count on each side. Record any revision separately.
6. **Exit · 3 min.** Submit exact work. [Key](ASSESSMENT.md#day-5-check-a): `v=4`, $32 each; at 3 D lower, at 6 C lower.

## Day 6 — Who is the population?

**Success:** distinguish a full group from a sample and note practical/ethical limits. (`AC9M8ST01`, `AC9M8ST04`)

1. **Frame · 2 min.** Read fictional question: “What travel modes occur in the imagined 100-student population?” No real classmates are surveyed.
2. **Model · 4 min.** Define population 100, census all 100, sample fewer. A census gives full counts in this model but can be costly or intrusive in a real school.
3. **Guided classify · 5 min.** Learners label A/B as constructed simple random samples of 20 and C as convenience sample of 20 by racks.
4. **Independent plan · 7 min.** Write a way to select 20 from numbered fictional cards giving each one a known equal chance, without replacing selected cards. State one privacy question a real survey would need to resolve.
5. **Peer audit · 4 min.** Partner checks whether selection location favours a mode and whether all 100 could be reached by the proposed method.
6. **Exit · 3 min.** “Does 20 responses automatically mean a fair sample?” Key: no; how people were selected matters.

## Day 7 — Same size, different results

**Success:** calculate proportions and describe variation between two same-size model random samples. (`AC9M8ST02`, `AC9M8ST03`)

1. **Recall · 2 min.** Both A and B have 20 responses. Ask whether identical size requires identical counts.
2. **Model · 4 min.** Bicycle share A is `3/20=15%`; B is `4/20=20%`.
3. **Guided table · 5 min.** Compute bus shares A `7/20=35%`, B `5/20=25%`; ask which category changed more in percentage points.
4. **Independent comparison · 7 min.** Learners choose walk/roll or other, compute both shares, and say why two model samples can differ even when each uses a fair chance process.
5. **Visual check · 4 min.** Show 20-cell grids or tactile tokens for each sample; prevent a 5-percentage-point difference being mistaken for five people.
6. **Exit · 3 min.** “How many percentage points apart are bicycle shares A and B?” Key: 5 points, one person out of 20.

## Day 8 — The location changed the answer

**Success:** critique a convenience sample without calling its arithmetic wrong. (`AC9M8ST01`, `AC9M8ST02`, `AC9M8ST04`)

1. **See · 2 min.** Display rack C: 16 bicycles out of 20; ask for its sample percentage.
2. **Model · 4 min.** `16/20=80%`, which is correct for C. Because selection was beside racks, it may overrepresent cyclists; it cannot estimate all students well.
3. **Guided compare · 5 min.** Contrast C's 80%, random model A's 15%, B's 20% and known fictional population's 20%. Ask which source answers each narrower question.
4. **Independent critique · 7 min.** Learners write a two-line correction to “80% of all students cycle”: valid sample result plus uncertainty about population.
5. **Fairness talk · 4 min.** Ask why a shelter decision based on the rack sample could overlook bus users. Do not ask anyone to disclose their own travel.
6. **Exit · 3 min.** “What is wrong: `16/20=80%` or calling that 80% of all students?” Key: the population inference.

## Day 9 — Report uncertainty, not a fake certainty

**Success:** make an inference limited by how the sample was chosen. (`AC9M8ST03`, `AC9M8ST04`)

1. **Recall · 2 min.** Ask why A and B's bicycle proportions differ by one person even though both have 20.
2. **Model · 4 min.** Say: “In these two simulated random samples, bicycle estimates were 15% and 20%; another fair sample could differ too.” Show the [optional repeated simulation](MATERIALS.md#week-2-invented-travel-population-and-samples): across 1000 seeds, the average error was 8.95 percentage points for samples of 10 and 3.92 for samples of 40. Do not claim every larger sample must be closer.
3. **Guided claim test · 5 min.** Sort “sample A had 3 cyclists” as direct model fact; “all students cycle at 15%” as overreach; “larger fair samples often vary less, but do not guarantee exactness” as a qualified pattern.
4. **Independent brief · 7 min.** Learners draft three sentences for a fictional principal: two source figures, one method limitation and one ethical next action. A source label must accompany each number.
5. **Peer check · 4 min.** Partner asks how a simple random sample would be drawn and whether a census is necessary or appropriate.
6. **Exit · 3 min.** “Can two equal-size random samples give different percentages?” Key: yes; this model has 15% and 20% bicycle.

## Day 10 — Fresh sample check B

**Success:** independently analyse a new sampling claim and propose a fairer response. Prepare [arts-club card](ASSESSMENT.md#day-10-check-b). (`AC9M8ST01`, `AC9M8ST02`, `AC9M8ST03`, `AC9M8ST04`)

1. **Set · 2 min.** Clarify fictional status and that no learner interest is being asked.
2. **Read · 4 min.** Give the 100-person model population and three 20-person samples. Do not name the likely biased sample.
3. **Plan · 5 min.** Learners decide how to compare percentages, selection method and the claim's reach.
4. **Independent work · 7 min.** Compute shares, identify sampling concern and write a bounded population statement.
5. **Check · 4 min.** Multiply or divide to confirm each fraction; label percentages as model sample or known model population.
6. **Exit · 3 min.** “Why not use the music-club meeting to speak for all 100?” Key: people there are more likely to be interested. Use [rubric/key](ASSESSMENT.md#day-10-check-b).

**Rights:** Original lesson prose © NeuroForgeIO Pty Ltd 2026, SubjectNest, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/); credit, link and indicate changes.