# Ten General Mathematics opening lessons — 25 minutes each

Use the [learner cards](LEARNER-CARDS.md), release the [fresh Day 5](DAY-05-CHECK.md) and [fresh Day 10](DAY-10-CHECK.md) cards only on those days, and use the paired [rate](../print/rate-table.svg) and [budget](../print/budget-map.svg) aids or [text equivalents](../print/TEXT-ALTERNATIVES.md). Keep the [teacher key](ASSESSMENT.md) separate. Each day totals **2 + 4 + 5 + 7 + 4 + 3 = 25 minutes**. This samples Unit 1 Topic 1 Consumer arithmetic (`GM-U1-T1`, [source ledger](../../../SOURCES-AND-REVIEW.md)); it is not the whole 14-hour topic. Every number is an invented teaching assumption, not a legal pay rate or financial recommendation.

## Day 1 — A rate needs a unit

**Success:** calculate a stated amount from hours × dollars per paid hour and label units.

1. **Notice · 2 min.** Display `$24 per paid hour`; ask whether 5 hours should give $29, $120 or another amount.
2. **Model · 4 min.** Teacher writes `5 paid h × $24/paid h = $120`; cancel hour units aloud and estimate five twenties ≈ $100.
3. **Guided trial · 5 min.** Calculate 3 hours = $72 and 7 hours = $168; ask why adding 24 once is insufficient.
4. **Independent card · 7 min.** Learners calculate 2, 5 and 6 base paid hours, each with expression, units and one reasonableness sentence. Calculator permitted after estimate.
5. **Check · 4 min.** Partner tests whether 6 hours is more than 5 by exactly $24 and corrects a missing unit label.
6. **Exit · 3 min.** Key: five hours $120; six hours $144; the difference is one $24 paid hour. “Paid” matters because breaks are unspecified.

## Day 2 — Multiplier versus total

**Success:** compute a fictional 1.25 multiplier without mistaking it for a $1.25 addition.

1. **Predict · 2 min.** Ask if Saturday model rate at 1.25 times $24 is above or below $24.
2. **Model · 4 min.** Split `1.25=1+0.25`: $24 + one quarter of $24 ($6) = $30 per hour.
3. **Guided · 5 min.** Calculate four Saturday paid hours = $120; compare five base hours = $120. Label equal totals, unequal hours/rates.
4. **Independent variation · 7 min.** Find Saturday totals for 2 and 6 paid hours; explain which operation applies before multiplying hours.
5. **Error audit · 4 min.** Discuss fabricated `$24+$1.25=$25.25`. Students explain why multiplier has no dollar unit and the answer should be $30.
6. **Exit · 3 min.** Key: 2 h=$60; 6 h=$180; 1.25×$24/h=$30/h under this fictional card.

## Day 3 — An extra-hour rule with a boundary

**Success:** apply the card's special rule to the specified shift only and avoid generalising it to real work.

1. **Frame · 2 min.** Read aloud “seven base hours, then two extra at 1.50× base” and point to the **fictional contract** label.
2. **Model · 4 min.** Calculate `$24×1.50=$36` per extra paid hour; `7×24=168`, `2×36=72`, combined $240.
3. **Guided compare · 5 min.** Ask why `9×36=$324` is wrong: only the final two of this example use $36. Check 8-hour version = $168+$36=$204.
4. **Independent boundary · 7 min.** Learners work a 10-hour what-if under the same stated rule: seven at base plus three extra. They show grouped arithmetic and label it as *this model*.
5. **Reasonableness · 4 min.** Partner checks adding one extra hour from nine to ten adds $36, not $24; teacher asks which term changes.
6. **Exit · 3 min.** Key: 10-hour model $168+$108=$276. No claim about an actual award or legal entitlement follows.

## Day 4 — Percent off is a change, not the final price

**Success:** distinguish discount amount and new price with a plausible check.

1. **Estimate · 2 min.** For fictional $80 item at 15% off, predict final price between $60 and $80.
2. **Model · 4 min.** `0.15×80=12`; new price `80−12=$68`. Compare `0.85×80=68` as a check.
3. **Guided · 5 min.** For $45 sign at 20% off, one fifth is $9; final $36. Explain that $9 is discount, not bill.
4. **Independent sort · 7 min.** Learners match old price, percent, dollar change and new price for both cards, then invent a **fictional** $100 item at 5% off and solve.
5. **Check · 4 min.** Partner audits whether new price is below old and above zero; teacher samples equations.
6. **Exit · 3 min.** Key: $80 item $12 discount/$68 new; $45 sign $9 discount/$36 new; $100 item $5/$95.

## Day 5 — Independent rate and percentage check A

**Success:** solve a fresh stipulated contract and explain every unit. Use [Day 5 check A](ASSESSMENT.md#day-5-check-a).

1. **Conditions · 2 min.** Give a new invented base `$28/h`, Saturday `1.25×`; ask for a rate and two totals. Calculator policy is stated.
2. **Read · 4 min.** Learners restate exactly what is known: 3 weekday paid hours; 4 Saturday paid hours; no break/travel terms.
3. **Plan · 5 min.** Individually select multiplication order and estimate before calculating. Teacher does not model this exact problem.
4. **Solve · 7 min.** Keep first work: Saturday rate, each shift total, difference and one unknown for a real comparison.
5. **Self-check · 4 min.** Recompute one route a second way; circle any number that has the wrong unit.
6. **Exit · 3 min.** Submit; key: $35/h, weekday $84, Saturday $140, difference $56. Record prompting and next move, not a QCAA grade.

## Day 6 — Unit price is useful, but not always the answer

**Success:** calculate per-sheet rate and compare whole-pack outlay at a specified need.

1. **Hook · 2 min.** Ask whether “cheaper per sheet” always means “spend less today”. Accept predictions.
2. **Model · 4 min.** From Paper Loop card: $18.40÷8=$2.30/sheet; $27÷12=$2.25/sheet. Large has lower unit price.
3. **Guided scenario · 5 min.** Need exactly eight; both whole packs cover it. Small outlay $18.40, large $27.00, difference $8.60; unused sheets are not free money.
4. **Independent compare · 7 min.** Learners solve twenty-four needed with whole packs: 3 small versus 2 large. Show pack counts and total, not just unit prices.
5. **Peer audit · 4 min.** Partner checks `24÷8=3`, `24÷12=2`, and whether delivery is known. Name the condition of unopened whole packs.
6. **Exit · 3 min.** Key: 24 sheets cost $55.20 small or $54.00 large; large is $1.20 lower in stated model.

## Day 7 — A percentage change can flip a comparison

**Success:** apply a rise to the correct base then redo a whole-pack choice.

1. **Retrieve · 2 min.** Learners state the original two-large total $54 and three-small total $55.20.
2. **Model · 4 min.** Large pack alone rises 10%: `$27×0.10=$2.70`; new pack $29.70; two $59.40.
3. **Guided compare · 5 min.** Keep small pack unchanged: three $55.20. Difference $4.20; the previously lower total is now higher. Clearly mark *what-if*.
4. **Independent check · 7 min.** Learners calculate a separate 8% rise on fictional $50 design fee = $54, and explain why they must not add 8 dollars.
5. **Challenge · 4 min.** Partners test whether 10% off $29.70 returns to $27; it gives $26.73. Discuss base changes; calculate $2.70/$29.70≈9.09% if ready.
6. **Exit · 3 min.** Key: after the 10% large-pack-only rise, three small packs are $4.20 lower for 24 sheets, before unknown delivery.

## Day 8 — Fixed costs and what remains

**Success:** build a fictional budget from a total, fixed costs and an unassigned balance.

1. **Read · 2 min.** Display the project fund $480; ask learners to distinguish money available from money already planned.
2. **Model · 4 min.** Add $120+$70+$45=$235; subtract `480−235=$245` unassigned. No personal finances are discussed.
3. **Guided add-on · 5 min.** If two large packs are selected for $54, remaining $191. If the $480 fund is unchanged, total planned cost is $289.
4. **Independent what-if · 7 min.** Learners choose either three small packs at $55.20 or two large at $54 for 24 sheets; calculate total planned and remaining, and say what cost is still unknown.
5. **Check · 4 min.** Partner verifies `planned + remaining = 480` and refuses to set unknown delivery to $0.
6. **Exit · 3 min.** Key: with small packs planned $290.20/remaining $189.80; with large $289/remaining $191.

## Day 9 — Spreadsheet as a transparent model

**Success:** enter a formula, change one assumption and explain the resulting difference. No spreadsheet is required if unavailable.

1. **Predict · 2 min.** What would six Saturday hours do to the original four-hour $120 result?
2. **Model · 4 min.** On a spreadsheet or board use Card 1 cells: `B1=24`, `B2=1.25`, `B4=4`; formula `=B1*B2*B4` displays 120. Name each cell's unit.
3. **Guided change · 5 min.** Change B4 to 6; formula gives $180. Difference $60 equals two more paid hours at $30/h.
4. **Independent replication · 7 min.** Learners make or draw a four-row table with base rate, multiplier, hours, total; test B4=2 and B4=5, record formulas and both outcomes.
5. **Audit · 4 min.** Partner changes one assumption and asks which formula reference updates. Recheck by direct multiplication to avoid trusting a mis-entered formula.
6. **Exit · 3 min.** Key: 2h=$60, 5h=$150. A spreadsheet output is a consequence of entered assumptions, not proof of real pay.

## Day 10 — Fresh consumer-arithmetic check B

**Success:** select a method for a new quantity/price comparison and articulate a missing condition. Use [Day 10 check B](ASSESSMENT.md#day-10-check-b).

1. **Set task · 2 min.** Present fictional Art Relay packs for a community display. Ask learners to choose whole packs for 30 badges, with no return of extras.
2. **Read · 4 min.** Fresh card: 10 badges/$22; 15 badges/$31.50. No delivery or storage cost supplied. Do not model the answer.
3. **Plan · 5 min.** Learners estimate unit prices, whole-pack counts and possible total before calculator/spreadsheet.
4. **Solve · 7 min.** Independently calculate unit prices and two ways to meet exactly 30, state lower given outlay and its size, then test a 10% rise of the 15-pack price.
5. **Self-audit · 4 min.** Check pack counts and units; distinguish the original comparison from the changed-price what-if.
6. **Exit · 3 min.** Submit first and checked work. Key: three 10-packs $66; two 15-packs $63, lower by $3; after 10% 15-pack rise, two cost $69.30 so 10-packs are $3.30 lower. Delivery remains unknown.

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