# Year 8 mathematics · ten 25-minute teacher scripts · Days 11–20

Use the [fictional source cards](MATERIALS.md), [clean learner page](LEARNER.md), [three daily practice routes](DAILY-CHOICES.md), [optional extras](DAILY-EXTRAS.md), [A4 aids and text/tactile routes](print/TEXT-ALTERNATIVES.md) and [separate staff key](teacher/ANSWER-AND-NEXT.md). All money and quantities are invented; no real purchase, learner household data or human outcome is being modelled. The [exact ACARA v9 crosswalk](CURRICULUM-CROSSWALK.md) records **partial** coverage. Speech, AAC, keyboard, tactile symbols, paper or a neutral scribe can carry the same mathematics. Distinguish access assistance from a mathematical hint.

**Ordinary rhythm:** launch 2 + model 5 + guided 6 + one A/B/C practice 6 + exit 4 + evidence note 2 = **25 minutes**. On Days 15/20 use launch 2 + prior example 5 + instructions 3 + fresh check 10 + transfer exit 3 + note 2 = **25 minutes**. The teacher may adjust local timing while protecting an independent first check response.

## Week 3 · expressions and equivalent forms

### Day 11 · Distribute to every part

**Codes:** `AC9M8A01`. **Goal:** expand a linear expression and prove equivalence with a substitution. **Prepare:** [expression tiles](print/expression-tiles.pdf) or labelled text strips.

1. **Launch · 2 min.** A fictional maker prepares three identical packs, each with `x` plain cards and 4 marker cards. Ask what `3(x+4)` counts. Name `x` before operating.
2. **Model · 5 min.** Draw three groups of `x+4`. Count `x+x+x+4+4+4=3x+12`. State the distributive property: `3(x+4)=3x+12`. At `x=2`, both give 18 cards. An example check supports, but does not alone prove all values; the grouping explains the identity.
3. **Guided · 6 min.** With four groups, expand `4(y+2)=4y+8`; substitute `y=3`: both 20. Read each multiplication sign aloud if useful.
4. **Practice · 6 min.** Offer D11-A/B/C from the [choice sheet](DAILY-CHOICES.md): tile array, spoken script or error hunt. Require distribution to **both** terms and a new-value check.
5. **Exit · 4 min.** Correct `2(z+6)=2z+6` and show at `z=1`: original 14 versus wrong 8; corrected `2z+12=14`.
6. **Note · 2 min.** Record property and substitution separately. If only one term was multiplied, return to two physical groups, not a speed drill.

**Later:** [Day11 extra](DAILY-EXTRAS.md) uses another invented pack, with no device needed.

### Day 12 · Collect like terms without losing signs

**Codes:** `AC9M8A01`. **Goal:** simplify a linear expression using commutative/associative grouping and check equivalence. **Prepare:** signed term strips.

1. **Launch · 2 min.** Ask which terms in `4x+7+2x−3` change with `x`, and which do not.
2. **Model · 5 min.** Regroup `(4x+2x)+(7−3)=6x+4`. At `x=2`, both forms are 16. The negative sign travels with the 3.
3. **Guided · 6 min.** Simplify `3a+9+a−5=4a+4`; check `a=1` gives 8 in both. Ask why `a` means `1a`, not zero.
4. **Practice · 6 min.** D12-A/B/C use different signed expressions; a table, oral grouping or wrong-step critique all require the original and simplified forms plus substitution.
5. **Exit · 4 min.** Evaluate `2n+5+3n−1` at `n=2` two ways. `5n+4=14`; original `4+5+6−1=14`.
6. **Note · 2 min.** If learner combines unlike terms, sort variable and constant strips. If sign error, keep the `−3` on one card.

**Later:** Optional [Day12 extra](DAILY-EXTRAS.md) adds a context, not a new rule.

### Day 13 · Factor out what is common

**Codes:** `AC9M8A01`. **Goal:** reverse distribution and explain why two forms match. **Prepare:** [expression tile mat](print/expression-tiles.pdf).

1. **Launch · 2 min.** Write `6x+12`. Ask what appears in six equal groups.
2. **Model · 5 min.** Six `x` tiles and twelve unit tiles can be shared into six equal groups of `x+2`: `6x+12=6(x+2)`. Expand back. At `x=3`, both are 30.
3. **Guided · 6 min.** `4p+20=4(p+5)`; expand to verify. At `p=2`, both equal 28. Show a common **numerical** factor; do not call `4(p+20)` equivalent.
4. **Practice · 6 min.** D13-A/B/C choose grouping, narrated expansion or factorisation critique. Each route shows a factor and an inverse expansion.
5. **Exit · 4 min.** Complete `3t+15=3(t+__)`: blank 5. Verify with `t=1`: 18 on each side.
6. **Note · 2 min.** If learner identifies a factor but cannot verify, expand their brackets aloud and compare coefficients.

**Later:** [Day13 extra](DAILY-EXTRAS.md) explores more than one common factor without making greatest-factor fluency a hidden prerequisite.

### Day 14 · Rearrange a model to find the input

**Codes:** `AC9M8A01`, `AC9M8A03`. **Goal:** rearrange `C=8+5n`, interpret `n`, and review a model assumption. **Prepare:** [balance/rearrange mat](print/balance-rearrange.pdf).

1. **Launch · 2 min.** A fictional club kit has an 8-token setup and 5 tokens per identical refill. `n` is a non-negative whole number of refills. Ask what `C` measures.
2. **Model · 5 min.** From `C=8+5n`, subtract 8, divide by 5: `n=(C−8)/5`. For `C=33`, `n=5`; check `8+5×5=33`. Not every `C` produces a valid whole refill count.
3. **Guided · 6 min.** Given `C=23`, use the same form to get `n=3`. At `C=24`, calculation gives `16/5=3.2`, outside this whole-refill model; discuss a possible unmodelled fee, not fractional refills.
4. **Practice · 6 min.** D14-A/B/C use different fictional linear models; each finds an input, checks by substitution and states a domain/omitted-condition limit.
5. **Exit · 4 min.** In `C=8+5n`, what would `C=8` mean? `n=0` refills; only setup tokens. A real club may have other conditions.
6. **Note · 2 min.** If inverse order is swapped, substitute the candidate in the original. Keep algebra and model interpretation as separate evidence.

**Later:** [Day14 extra](DAILY-EXTRAS.md) tests a total that cannot represent a whole refill.

### Day 15 · Fresh expression transfer check

**Codes:** `AC9M8A01`, `AC9M8A03`. **Goal:** independently expand, simplify, factorise and invert a new fictional model. **Prepare:** release **only** the Day15 section of [learner checks](STUDENT-CHECKS.md); hold the [staff key](teacher/ANSWER-AND-NEXT.md).

1. **Launch · 2 min.** Say this is a new situation and an internal formative check. Neutral reading and accessible notation are allowed.
2. **Prior example · 5 min.** Revisit different `3(x+4)=3x+12` and the `x=2` check. Put it away before new numbers appear.
3. **Instructions · 3 min.** Ask learners to show every transformation and substitute back. Do not expand or factor the new expression for them.
4. **Fresh check · 10 min.** Give new-case Items 1–2. Save the first response; content prompts are noted separately.
5. **Transfer exit · 3 min.** Give Item 3, asking for a bounded interpretation. Later D15 choices use **different** values.
6. **Note · 2 min.** Separate distributive reasoning, collecting terms, inverse step and contextual limit. If access prevented response, mark `not observed` rather than wrong.

**Later:** Use D15 practice choices only **after** the check is secure.

## Week 4 · equations, inequalities and verification

### Day 16 · Keep an equation balanced

**Codes:** `AC9M8A02`. **Goal:** solve a two-step linear equation, including a rational solution in practice, and verify by substitution. **Prepare:** [balance/rearrange mat](print/balance-rearrange.pdf).

1. **Launch · 2 min.** Ask what equality means in `3x+7=25`: both sides hold the same value.
2. **Model · 5 min.** Subtract 7 on both sides, divide both sides by 3: `x=6`. Substitute: `3×6+7=25`. The balance picture helps explain inverse operations.
3. **Guided · 6 min.** Solve `2y+5=12`: `2y=7`, `y=7/2=3.5`; substitute `2×3.5+5=12`. A rational answer is legitimate when the variable has no whole-number domain restriction.
4. **Practice · 6 min.** D16-A/B/C solve distinct rational-result equations, with tiles, spoken balancing or error critique. Require exact fraction/decimal and substitution.
5. **Exit · 4 min.** In `4z−3=9`, `z=3`; check `4×3−3=9`. Ask why adding 3 happens before dividing 4.
6. **Note · 2 min.** If a learner alters one side only, use the balance mat; if arithmetic slips, preserve algebra evidence and check with calculator after setup.

**Later:** [Day16 extra](DAILY-EXTRAS.md) uses a negative starting constant.

### Day 17 · A variable can appear on both sides

**Codes:** `AC9M8A02`, `AC9M8A03`. **Goal:** solve and graph the crossing of two fictional linear plans, then state what the equality means. **Prepare:** [graph and number-line board](print/line-and-boundary.pdf).

1. **Launch · 2 min.** Fictional design plans: A `6+4n` tokens, B `18+2n` tokens for `n` whole batches. Ask what is fixed and what grows.
2. **Model · 5 min.** Set `6+4n=18+2n`. Subtract `2n`, then 6: `2n=12`, so `n=6`. Check both equal 30. At `n=5`, A 26/B 28; at 7, A 34/B 32.
3. **Guided · 6 min.** Plot A's `(0,6),(6,30)` and B's `(0,18),(6,30)` with labelled axes. The intersection agrees with algebra; integer x-values are the actual batch cases. Draw a line for shape but do not interpret half a batch as offered.
4. **Practice · 6 min.** D17-A/B/C use new pairs; each route finds equality algebraically, verifies and sketches or describes the crossing.
5. **Exit · 4 min.** Which plan is lower at seven batches and by how much? B=32 versus A=34, so B by 2 tokens under this invented model.
6. **Note · 2 min.** If graph/intersection is approximate, use substitution to settle the exact equality. Record unit and omitted conditions.

**Later:** [Day17 extra](DAILY-EXTRAS.md) checks a nearby count.

### Day 18 · An inequality describes many allowed inputs

**Codes:** `AC9M8A02`, `AC9M8A03`. **Goal:** solve a one-variable inequality and interpret its whole-number boundary. **Prepare:** [graph/number-line board](print/line-and-boundary.pdf).

1. **Launch · 2 min.** A fictional poster desk has 9 setup points and 5 points per poster, with 34 points available. Let `n` be whole posters, `n≥0`.
2. **Model · 5 min.** `9+5n≤34` means within cap. Subtract 9, divide by positive 5: `n≤5`. Allowed whole counts are 0–5. Check boundary: at 5, 34; at 6, 39 and over cap.
3. **Guided · 6 min.** Draw closed dot at 5 with arrow left on number line; cross out negative values by domain. On Cartesian axes, show `y=9+5n` and horizontal `y=34`; eligible integer points are at/below budget line.
4. **Practice · 6 min.** D18-A/B/C: new cap and relation, each with algebra, two checks and a number-line/graph or precise verbal equivalent.
5. **Exit · 4 min.** Explain the difference between `≤` and `<` at the boundary. With `n=5`, `9+5n≤34` true; `9+5n<34` false.
6. **Note · 2 min.** If learner says only `n=5`, have them test `n=4` and `n=0`; record solution set, not just maximum.

**Later:** [Day18 extra](DAILY-EXTRAS.md) uses an exclusive cap.

### Day 19 · Negative multiplication reverses the order

**Codes:** `AC9M8A02`. **Goal:** solve an inequality with a negative coefficient and verify boundary/nearby values. **Prepare:** [graph/number-line board](print/line-and-boundary.pdf).

1. **Launch · 2 min.** Consider a purely fictional counter tray that starts at 15 and loses 3 counters each turn: `r=15−3t`, `t=0,1,2,3,4,5`. Ask when more than 3 remain.
2. **Model · 5 min.** `15−3t>3`; subtract 15: `−3t>−12`; divide by `−3` **and reverse** sign: `t<4`. Allowed turns are 0,1,2,3. Test 3→6 (true), 4→3 (not greater).
3. **Guided · 6 min.** Show why the reversal is necessary by testing `t=5`: `15−15=0`, so `t>4` would be wrong. Draw an open dot at 4, arrow left, then restrict to the stated whole-number domain.
4. **Practice · 6 min.** D19-A/B/C solve other negative-coefficient inequalities, check boundary and a value on each side, and explain reversal in words.
5. **Exit · 4 min.** Solve `8−2u≥0`: `u≤4`, with `u=4` true at zero and `u=5` false. If no physical context is specified, give the real-number set; if whole turns, list non-negative whole values through 4.
6. **Note · 2 min.** If sign stays unchanged after dividing negative, test a counterexample before re-teaching the symbolic rule.

**Later:** [Day19 extra](DAILY-EXTRAS.md) practises another reversal with a different sign.

### Day 20 · Fresh equation and inequality transfer check

**Codes:** `AC9M8A02`, `AC9M8A03`. **Goal:** build and verify an equality plus a constrained inequality in a new fictional batch situation. **Prepare:** new-case Day20 [learner check](STUDENT-CHECKS.md) and public teacher [worked key](teacher/ANSWER-AND-NEXT.md).

1. **Launch · 2 min.** Explain the first work tells us what to teach next; the scenario is not a real procurement decision.
2. **Prior example · 5 min.** Revisit different Day17 plans at `n=6` and the Day18 `n≤5` boundary check. Remove example figures before check.
3. **Instructions · 3 min.** Ask for an equation, inequality, exact solutions and substitution. Offer a blank balance/graph board without numbers.
4. **Fresh check · 10 min.** Give new-case Items 1–2. Allow neutral reading, AAC, large print or tactile symbols. Record content hints separately.
5. **Transfer exit · 3 min.** Item 3 asks for an interpretation with a model limit, not a shopping recommendation. Save first response before coaching.
6. **Note · 2 min.** Mark equations, inequality direction, graphical relation, verification and context independently. Use later D20 choices with **different** numbers.

**Later:** [Day20 extra](DAILY-EXTRAS.md) is only for practice after the held-out response.

Original SubjectNest teacher scripts © NeuroForgeIO Pty Ltd 2026, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/).