Mathematics · ten 25-minute… | Year 8 Maths · T1 W3–4 · Lessons

Use the fictional source cards, clean learner page, three daily practice routes, optional extras, A4 aids and text/tactile routes and separate staff key. All money and quantities are invented; no real purchase, learner household data or human outcome is being modelled. The exact ACARA v9 crosswalk 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 or labelled text strips.

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.
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.
Guided · 6 min. With four groups, expand 4(y+2)=4y+8; substitute y=3: both 20. Read each multiplication sign aloud if useful.
Practice · 6 min. Offer D11-A/B/C from the choice sheet: tile array, spoken script or error hunt. Require distribution to both terms and a new-value check.
Exit · 4 min. Correct 2(z+6)=2z+6 and show at z=1: original 14 versus wrong 8; corrected 2z+12=14.
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 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.

Launch · 2 min. Ask which terms in 4x+7+2x−3 change with x, and which do not.
Model · 5 min. Regroup (4x+2x)+(7−3)=6x+4. At x=2, both forms are 16. The negative sign travels with the 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.
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.
Exit · 4 min. Evaluate 2n+5+3n−1 at n=2 two ways. 5n+4=14; original 4+5+6−1=14.
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 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.

Launch · 2 min. Write 6x+12. Ask what appears in six equal groups.
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.
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.
Practice · 6 min. D13-A/B/C choose grouping, narrated expansion or factorisation critique. Each route shows a factor and an inverse expansion.
Exit · 4 min. Complete 3t+15=3(t+__): blank 5. Verify with t=1: 18 on each side.
Note · 2 min. If learner identifies a factor but cannot verify, expand their brackets aloud and compare coefficients.

Later: Day13 extra 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.

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.
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.
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.
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.
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.
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 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; hold the staff key.

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

Launch · 2 min. Ask what equality means in 3x+7=25: both sides hold the same value.
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.
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.
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.
Exit · 4 min. In 4z−3=9, z=3; check 4×3−3=9. Ask why adding 3 happens before dividing 4.
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 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.

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.
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.
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.
Practice · 6 min. D17-A/B/C use new pairs; each route finds equality algebraically, verifies and sketches or describes the crossing.
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.
Note · 2 min. If graph/intersection is approximate, use substitution to settle the exact equality. Record unit and omitted conditions.

Later: Day17 extra 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.

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.
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.
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.
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.
Exit · 4 min. Explain the difference between ≤ and < at the boundary. With n=5, 9+5n≤34 true; 9+5n<34 false.
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 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.

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.
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).
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.
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.
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.
Note · 2 min. If sign stays unchanged after dividing negative, test a counterexample before re-teaching the symbolic rule.

Later: Day19 extra 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 and public teacher worked key.

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

Later: Day20 extra is only for practice after the held-out response.