# Ten daily mathematics lessons — Days 1–10

**Every day: 25 minutes = 3 retrieve/notice + 5 explicit model + 8 guided practice + 6 independent transfer + 3 exit and record.** [Context/key cards](aids.md), [A4 coordinate grid](coordinate-grid.svg), [blank model plot](line-comparison.svg) and [assessment](assessment.md) are original. Keep a calculator available for arithmetic access, but ask for the relation and units. Give a large-print/tactile/digital grid or narrated coordinates as needed. If an adult places a point under learner direction, score the learner's decision and note the access route. All contexts are invented; no physical safety or real price conclusion follows. AC9M9A03 is the principal Weeks 1–2 link; AC9M9A05 is a **partial** Week 2 link; see [scope](README.md).

## Day 1 — Two points, two signed changes

**Goal/materials:** recover ordered pairs and calculate horizontal/vertical change; grid, Card 1, two markers. This is a readiness check, not a speed test.

1. **Notice, 3:** Mark A(-4,1) and B(2,4). Ask “Which coordinate tells us across?” **Key:** x. If x/y are swapped, trace axes labels before proceeding.
2. **Model, 5:** Say “From A to B I move 6 right: 2−(-4)=6. I move 3 up: 4−1=3. These are changes, not the endpoint coordinates.” Mark a right-angle path on the grid.
3. **Together, 8:** Plot Card 2 (1,4)→(5,0); calculate Δx=+4 and Δy=-4. Repeat with points reversed, getting Δx=-4 and Δy=+4. Discuss that a direction change reverses both signs.
4. **Try, 6:** New fictional game route (-3,-2)→(2,1). Learner plots and states Δx, Δy in speech, text, tactile steps or AAC. **Key:** +5,+3. Teacher records coordinate/negative-number prerequisite separately.
5. **Check, 3:** “From (4,-1) to (1,3), what changed?” **Key:** x=-3, y=+4. If learner subtracts in mixed order, write end minus start in two columns, then fresh recheck.

## Day 2 — Gradient is rise divided by run

**Goal/materials:** form and interpret gradient when run is nonzero; grid and Cards 1–2.

1. **Notice, 3:** Retrieve Card 1 Δy=3, Δx=6. Ask which describes vertical change **per** horizontal unit. **Key:** 3/6.
2. **Model, 5:** “Gradient m=rise/run=3/6=1/2. On this paper grid, moving 2 right rises 1. The axes use equal grid units; I have not measured a real-world slope.” Draw one step triangle.
3. **Together, 8:** Card 2 Δy=-4, Δx=4 ⇒ m=-1. Trace 1 right/1 down. Then reverse point order: (-Δy)/(-Δx)=-1; gradient stays the same for the same line.
4. **Try, 6:** New theatre-map points (-2,0)→(4,3). **Key:** Δx=6, Δy=3, m=1/2. Learner labels rise/run; calculator can simplify fraction.
5. **Check, 3:** “What is m from (0,5) to (3,-1)?” **Key:** -6/3=-2. If answer +2, attend to the downward change with a signed arrow, then recheck new points.

## Day 3 — Zero and undefined gradients are different

**Goal/materials:** distinguish horizontal zero gradient from vertical undefined gradient; grid, Cards 3–4.

1. **Notice, 3:** Ask what happens to y on Card 4 as x changes. **Key:** y remains -2.
2. **Model, 5:** “Card 4 has Δy=0, Δx=8, so m=0/8=0. Card 3 has Δx=0, so a rise/run fraction would divide by zero; its gradient is **undefined**, not zero.” Use perpendicular paper strips.
3. **Together, 8:** Sort (0,2)→(5,2), (3,-1)→(3,4), (-2,-2)→(2,2). **Keys:** zero, undefined, +1. Explain before applying a label.
4. **Try, 6:** Learner creates one new horizontal and one new vertical pair inside grid limits, gives reasons and marks them. **Accept:** any distinct points sharing y or sharing x respectively.
5. **Check, 3:** “Gradient from (-4,3) to (-4,-2)?” **Key:** undefined, because Δx=0. If called 0, contrast one horizontal and one vertical pair with explicit denominators.

## Day 4 — The midpoint averages each coordinate

**Goal/materials:** locate halfway point even with negative coordinates; grid, Cards 5–6.

1. **Notice, 3:** On a line, what is halfway between -6 and +4? **Key:** -1. Check equal distance 5 each way.
2. **Model, 5:** Card 5 A(-6,2), B(4,8). “Average x values: (-6+4)/2=-1. Average y: (2+8)/2=5. Midpoint (-1,5).” Plot and check equal horizontal/vertical moves.
3. **Together, 8:** Card 6 (-3,-4),(5,2) ⇒ M(1,-1). Swap point order to show midpoint invariant. Check x displacements 4/4, y 3/3.
4. **Try, 6:** Fresh design segment (-5,-1)→(3,7). **Key:** M(-1,3). Learner may state/plot and justify equal displacement.
5. **Check, 3:** “Midpoint of (1,5) and (7,-1)?” **Key:** (4,2). If coordinate pairs are cross-averaged, colour/tactile-code x with x and y with y, then try a new pair.

## Day 5 — Straight-line distance is not a route length

**Goal/materials:** use Pythagoras for coordinate distance; grid, Cards 7–9, calculator/square table available.

1. **Notice, 3:** On Card 7 (0,0)→(6,8), mark horizontal 6 and vertical 8. Ask whether their sum 14 is the straight shortcut. **Key:** no, 14 is the two-leg path.
2. **Model, 5:** “The right triangle gives d²=6²+8²=36+64=100, so d=10 **grid units**.” Do not call this a travel time, accessible route or device measurement.
3. **Together, 8:** Card 8 (-2,1)→(1,5): legs 3 and 4, d=5. Card 9 (1,-2)→(7,6): legs 6 and 8, d=10. Check the distance is positive and unchanged if point order swaps.
4. **Try, 6:** Fresh park-plan sketch (-1,-1)→(2,3). **Key:** Δx=3, Δy=4, d=5 grid units. Learner shows squared legs or a right triangle.
5. **Check, 3:** “Distance between (3,2) and (3,-5)?” **Key:** 7 units. If learner squares the vertical difference unnecessarily, show direct axis interval and compare the general formula. [Assessment](assessment.md) Items 1–4 may be collected across Days 4–6.

## Day 6 — Choose the right relationship

**Goal/materials:** select gradient, midpoint or distance from a question; Cards 10–12, grid. Begin with a three-item prerequisite pulse before new model.

1. **Notice, 3:** Point to axes; ask “Which question wants steepness, halfway, straight separation?” **Keys:** gradient, midpoint, distance. Record any category confusion.
2. **Model, 5:** Card 11 (-6,-4)→(2,2). “For steepness Δy/Δx=6/8=3/4. For straight separation √(8²+6²)=10. Same points, different question and unit.”
3. **Together, 8:** Card 10 (-5,3)→(-1,-3): m=-6/4=-3/2 and M(-3,0). Card 12 (-4,-6)→(4,0): M(0,-3), d=10. Ask why midpoint cannot answer distance.
4. **Try, 6:** Fresh stage layout (-4,0)→(2,8): find midpoint and distance. **Keys:** (-1,4), √(6²+8²)=10. Mark mathematical prompts separately from graph access.
5. **Check, 3:** “From (0,0) to (4,2), which number is the gradient and which point is the midpoint?” **Keys:** 1/2; (2,1). If measures conflated, attach units/point notation to the requested output.

## Day 7 — A gradient becomes a rate in context

**Goal/materials:** derive C(h)=12+6h from two invented plot points; [blank plot](line-comparison.svg), graph-check card.

1. **Notice, 3:** Show fictional print-studio points (0,12),(4,36) with h in whole booked hours and C in credits/dollars. Ask for vertical change. **Key:** 24 credits.
2. **Model, 5:** “Rate m=24/4=6 credits per hour; at h=0, cost is 12. Model C(h)=12+6h for 0–8 whole hours. This is a stipulated quote, not a live offer.” Plot points (0,12),(4,36),(8,60).
3. **Together, 8:** Groups derive D(h) from (0,24),(4,40): rate 4, fixed 24. Plot (0,24),(4,40),(8,56). Name what the intercept and gradient mean in this fictional model.
4. **Try, 6:** New bike-locker simulation E(h) through (0,5),(3,20). **Key:** rate 5 credits/hour, fixed 5, E(h)=5+5h; E(4)=25. Learner states domain if teacher stipulates h=0–6 whole hours.
5. **Check, 3:** “Does a graph point at h=2.5 belong to our print-studio booking rule?” **Key:** no, booking domain is whole hours; a mathematical line can be drawn between points but a half-hour quote is not provided. If overlooked, re-read domain card.

## Day 8 — Compare two lines and a break-even point

**Goal/materials:** solve/interprete intersection and conditional advantage; C and D model card, plot and calculator.

1. **Notice, 3:** Retrieve C(0)=12 and D(0)=24. **Key:** C lower at 0 whole hours under stated model.
2. **Model, 5:** “Set 12+6h=24+4h. Subtract 4h and 12: 2h=12 ⇒ h=6. Both cost 48 at h=6.” Check by substituting both rules and matching graph intersection.
3. **Together, 8:** At h=3, C=30, D=36; at h=8, C=60, D=56. Ask which is lower for each and why the steeper C line becomes higher after 6. Avoid ‘better for everyone’.
4. **Try, 6:** Fresh training-room simulator F(h)=8+7h and G(h)=20+4h, h whole 0–7. **Key:** equal at h=4, both 36; at h=2, F22 vs G28, F lower. Learner checks algebra and context.
5. **Check, 3:** “Which of C or D is lower at h=6?” **Key:** neither; equal at 48. If learner uses only starting charge, plot/compute h=6 and say ‘depends on h’.

## Day 9 — A model has a domain and limits

**Goal/materials:** evaluate what a linear quote can and cannot justify; C/D card, optional graph and model-audit strip.

1. **Notice, 3:** Ask whether C(10) is a supported price. **Key:** arithmetic gives 72, but stated domain ends at 8; the pack does not supply a real quote for 10.
2. **Model, 5:** Teacher says “Within 0–8 whole hours, the rule compares stated charges. It omits availability, print quality, taxes and any extra fee. ‘D is always cheaper’ fails at h=3.” Show model → interpretation → check of assumption.
3. **Together, 8:** Consider three claims: “C cheaper at 3” (supported); “D cheaper at 8” (supported); “D prints better quality” (unknown). Learners identify the exact number or missing evidence behind each.
4. **Try, 6:** Fresh concert-prop hire H(t)=18+2t, J(t)=6+5t for t=0–6 whole days. **Keys:** equal when 18+2t=6+5t ⇒ t=4, cost 26; H lower at day 6 (30 vs 36); J lower at day 2 (16 vs 22). Learner names one omitted condition.
5. **Check, 3:** “If a real supplier adds a delivery fee, what must we do?” **Key:** revise the cost model using sourced terms; we cannot assume the given graph still applies. If learner answers only ‘choose cheap line’, revisit assumption card.

## Day 10 — Unseen transfer and next teaching

**Goal/materials:** independently choose coordinate tools and critique a new linear model; [assessment](assessment.md), grid and plot. Keep assessment figures unseen until this day.

1. **Notice, 3:** Retrieve `Δx, Δy → gradient`, `coordinate averages → midpoint`, `Pythagoras → distance`, and `fixed + rate × input → model`. Show symbols, not assessment numbers.
2. **Model, 5:** Correct an unscored error: from (0,1) to (2,5), “gradient 4/2=2, not 4; the rise is 4 but the run is 2.” Also name grid units.
3. **Together, 8:** Audit a new unscored line K(h)=9+3h: starting amount 9, rate 3 per hour, K(2)=15; compare its claim “always cheapest” with absent rival data. Use response routes before independent work.
4. **Try, 6:** Learner begins unseen [two-week assessment](assessment.md), Items 1–2 or an equivalent scheduled conference. Teacher may scribe only learner-directed steps and notes mathematical prompting separately.
5. **Check, 3:** Ask “Which assumption would you verify before using a classroom model to spend real money?” **Accept:** real prices/domain/fees or availability; no fictitious quote can justify a purchase. Complete remaining assessment in a separate 10–12-minute check, then select next lesson from rubric, not a single score.

## Shared teaching response

If a learner treats Δy as a gradient, show two lines with the same rise but different runs. If the denominator is zero, say “undefined” and trace the vertical line; never make it zero to fit the formula. If distance becomes |Δx|+|Δy|, name that as a two-leg path and contrast straight separation. If a learner produces algebra without context, require a unit, a domain and an assumption. If the concept is secure with a supported response route, remove only the mathematical prompt, not necessary access.

**Rights:** Original lessons © NeuroForgeIO Pty Ltd 2026, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). ACARA source and separate rights in [README](README.md). Educator/local review pending.