# Year 7 mathematics · Term 1 Weeks 3–4 · Days 11–20

**Each day is 25 minutes:** launch 2 + model 5 + guide 6 + one [practice choice](DAILY-CHOICES.md) 6 + exit 4 + note 2. On Days 15/20, the ten minutes for practice/exit become a **fresh check**; those days' three routes are later practice. Use [clean learner prompts](LEARNER.md), original [print/text aids](print/TEXT-ALTERNATIVES.md), and a [private staff key](teacher/ANSWER-AND-NEXT.md). Record exact mathematics, representation, access route and any content hint separately. A read-aloud can preserve mathematical reasoning without establishing independent reading.

## Week 3 · Choose and check operations with positive rational numbers

The first fortnight introduced equivalent rational forms, rounding, fraction addition/subtraction, decimal money arithmetic and simple percentages. This week **revisits and extends** these, adding multiplication/division. The physical whole or unit must be named. Every numeric situation below is invented for teaching; no purchase or personal finance disclosure is required.

### Day 11 · Add fractions without changing the whole

**Codes:** `AC9M7N04`, `AC9M7N06`. **Goal:** explain `1/2+3/8` and `5/6−1/3` with common units. **Prepare:** [operation and check mat](print/operation-check.pdf), matching fraction strips.

1. **Launch · 2 min.** Ask if half a fictional page plus three eighths of the **same** page is above or below one whole.
2. **Model · 5 min.** Rename `1/2=4/8`; then `4/8+3/8=7/8`, below one. Show the same whole strip, not two differently sized pages.
3. **Guide · 6 min.** Rename `1/3=2/6`; `5/6−2/6=3/6=1/2`. Learners explain why adding/subtracting denominators would change unit size. Compare result with zero and one.
4. **Choice · 6 min.** D11-A/B/C: original mosaic, time-segment or error-repair route. Every route needs an equivalent-name step and plausibility check.
5. **Exit · 4 min.** Solve `3/4−1/8=6/8−1/8=5/8`, then say why it is less than 3/4. This revisits a familiar operation to separate method from new numbers.
6. **Note · 2 min.** Record whether the learner preserved one whole and justified denominator choice; recheck a fresh pair if not.

**Optional home/extension:** Draw a same-length strip and solve `1/2+1/4=3/4`. Extension: two strategies for `5/6−1/3`.

### Day 12 · Decimal operations and a reasonable estimate

**Codes:** `AC9M7N05`, `AC9M7N06`. **Goal:** add and subtract decimals by place value, then check scale. **Prepare:** operation mat, decimal place cards.

1. **Launch · 2 min.** Two fictional design lengths are 4.75 m and 2.60 m. Expect a total near 7–8 m, not 70 m.
2. **Model · 5 min.** Align decimal places: `4.75+2.60=7.35 m`. Round each input to the nearest whole metre (`4.75→5`, `2.60→3`) for a loose estimate `5+3=8 m`; this detects a decimal-shift error but does not replace the exact sum.
3. **Guide · 6 min.** From a fictional 10.00 m roll, subtract 7.35 m: `10.00−7.35=2.65 m`. Check `7.35+2.65=10.00`. Ask why metres cannot be silently changed to centimetres.
4. **Choice · 6 min.** D12-A/B/C uses a fictional art strip, data-storage readout or sports path; each shows written/inverse check and unit.
5. **Exit · 4 min.** Repair `4.75+2.60=6.135`: show correct `7.35` and one magnitude reason. Do not credit a close estimate as the exact answer.
6. **Note · 2 min.** Save original decimal alignment and inverse. If error is in place value, use a different tenths/hundredths example next.

**Optional home/extension:** Use paper decimal columns; extension: express 2.65 m as 265 cm with the unit conversion stated.

### Day 13 · Multiply fractions as part of a part

**Codes:** `AC9M7N04`, `AC9M7N06`. **Goal:** explain positive fraction multiplication without a memorised rule alone. **Prepare:** same-whole rectangle/grid or raised partitions.

1. **Launch · 2 min.** In a fictional graphic, what should `3/4 of 2/3` be compared with `2/3`: larger or smaller? **Smaller**.
2. **Model · 5 min.** Partition one whole into thirds one way and quarters the other. The overlap for `3/4×2/3` is `6/12=1/2`. Explain the one-whole area model and then numerator/denominator product.
3. **Guide · 6 min.** Use `2/5×3/4=6/20=3/10`. Confirm `3/10<3/4` and `3/10<2/5` because both factors are below 1.
4. **Choice · 6 min.** D13-A/B/C: page panel, signal interval or rectangular grid. Learners represent an overlap and exact simplified result.
5. **Exit · 4 min.** Is `1/2×3/4` equal to `3/8` or `4/6`? **3/8**; explain with an eight-cell area or a half of three quarters.
6. **Note · 2 min.** Record whether learner understood “of” as a second partition of the same whole; recheck a new area model if only a rule was repeated.

**Optional home/extension:** Draw a 4-by-3 blank rectangle. Extension: explain why `3/4×2/3=2/3×3/4` by rotated partitions.

### Day 14 · Division and percentage are operation choices

**Codes:** `AC9M7N05`, `AC9M7N06`. **Goal:** identify how many 0.6-unit lengths fit in 2.4 units and find a familiar percentage. **Prepare:** number-line/decimal cards, operation mat.

1. **Launch · 2 min.** A fictional path is 2.4 units long; markers are 0.6 units apart. Predict roughly four intervals.
2. **Model · 5 min.** `2.4÷0.6=4`; multiply both quantities by 10 to use `24÷6`, then check `4×0.6=2.4`. Keep the quotient unit as **intervals**, not 4 units of length.
3. **Guide · 6 min.** Another fictional 1.5-unit strip in quarter-unit pieces: `1.5÷0.25=6` pieces; check six quarters =1.5. Then `12.5%=1/8`, so 12.5% of 48 equal panels is **6**. These are two operation types, with the whole named in each.
4. **Choice · 6 min.** D14-A/B/C gives a repeated-decimal grouping, percentage partition or operation-choice explanation; require the unit and an inverse/benchmark check.
5. **Exit · 4 min.** “Is `2.4÷0.6` equal to 0.4?” **No**; four groups of 0.6 fill 2.4. A quotient smaller than one cannot count four visible groups.
6. **Note · 2 min.** Separate grouping-division reasoning from percentage-of-whole reasoning; plan the missing strand as a next move.

**Optional home/extension:** Draw six equal quarter lengths to cover 1.5; extension: compare `1.5÷0.25` with `1.5×0.25` and explain why they differ.

### Day 15 · Fresh rational-operation check

**Codes:** `AC9M7N05`, `AC9M7N06`. **Goal:** transfer to unseen `2/3+1/4`, `7.2÷0.3` and 15% of 80. **Prepare:** closed [learner check](STUDENT-CHECKS.md), separate [staff key](teacher/ANSWER-AND-NEXT.md). Do not show check figures in routes before this point.

1. **Launch · 2 min.** Explain that a new problem shows which strategy to teach next, not a rank of students.
2. **Model · 5 min.** Rehearse different figures: `1/2+1/4=3/4` and `2.4÷0.6=4`; put them away.
3. **Guide · 6 min.** Find 25% of 40 =10 through one quarter. Ask what 100% names; do not show the held-out percent.
4. **Independent check · 6 min.** Give unseen Items 1–2. Read wording neutrally or supply a blank tactile representation without choosing steps.
5. **Check exit · 4 min.** Item 3 asks for a percentage of a named whole and a check. Preserve first responses; D15 routes are later practice.
6. **Note · 2 min.** Record equivalence, place value/grouping, percent whole and estimation separately. If access/time prevented a response, mark not observed.

**Optional home/extension:** Later practice only; do not coach the held-out page as homework.

## Week 4 · Prime products, powers and square roots

A prime factorisation uses **only primes**. Exponent notation compresses repeated multiplication: `2³=2×2×2=8`, not `2×3`. Distinct factor trees for one number must yield the same prime product. A positive perfect square has an integer square root; `√144=12` because `12²=144`, with the principal square root positive in this pack. The code links below are partial encounters, not broad mastery claims.

### Day 16 · Break a number into prime leaves

**Code:** `AC9M7N02`. **Goal:** factorise 60 and 84 with prime leaves and exponents. **Prepare:** [factor-tree mat](print/prime-factor-tree.pdf) or text/tactile card route.

1. **Launch · 2 min.** Ask why `6×10=60` is not yet a *prime* factorisation: 6 and 10 are composite.
2. **Model · 5 min.** Split 60 as `6×10=(2×3)×(2×5)=2×2×3×5=2²×3×5`. Check `4×3×5=60`.
3. **Guide · 6 min.** Factor 84 via `4×21=(2×2)×(3×7)=2²×3×7`; verify `4×3×7=84`. Other valid tree shapes are welcome.
4. **Choice · 6 min.** D16-A/B/C: tree, prime-leaf cards or repair of an unfinished product. Each ends with primes only and a multiplication check.
5. **Exit · 4 min.** Is `2²×3×5` equal to 60? **Yes.** Is `4×3×5` already a prime product? **No**, because 4 is composite.
6. **Note · 2 min.** Record where factorisation stopped and whether exponent notation matches leaf count.

**Optional home/extension:** Draw a 60 tree on paper. Extension: start 60 with `3×20` and compare final leaves.

### Day 17 · Read powers as repeated prime factors

**Code:** `AC9M7N02`. **Goal:** represent 72 accurately as `2³×3²` and distinguish base/exponent. **Prepare:** factor cards and tree.

1. **Launch · 2 min.** Ask what the small 3 in `2³` counts: **three factors of 2**.
2. **Model · 5 min.** Factor `72=8×9=(2×2×2)×(3×3)=2³×3²`. Check `8×9=72`; `2³` is 8, not 6.
3. **Guide · 6 min.** Start another tree `72=12×6=(2×2×3)×(2×3)`; regroup to `2³×3²`. A different branch order changes no prime counts.
4. **Choice · 6 min.** D17-A/B/C: tree comparison, spoken exponent description or error diagnosis with a new composite.
5. **Exit · 4 min.** Evaluate `2³×3²` as `8×9=72`, and expand `3²` as `3×3`.
6. **Note · 2 min.** If learner reads `2³` as 2×3, place three 2-cards and revisit with 3² as a contrasting card.

**Optional home/extension:** Make base/exponent cards; extension: compare `2²×3²=36` with `2³×3²=72` and explain the extra factor of 2.

### Day 18 · Prime factors can simplify a comparison

**Codes:** `AC9M7N02`, `AC9M7N04`. **Goal:** factorise 90 and 150, then explain a shared factor and equivalent fraction. **Prepare:** tree mat, blank ratio/fraction line.

1. **Launch · 2 min.** Two invented collections have 90 and 150 paper cards. Ask what common equal group size might simplify the comparison.
2. **Model · 5 min.** `90=2×3²×5`; `150=2×3×5²`. Both include `2×3×5=30`, so `90/150=(90÷30)/(150÷30)=3/5`. Verify `3/5=0.6`; do not infer that the collections have the same total.
3. **Guide · 6 min.** Draw two factor trees, then circle *one* matching 2, 3 and 5 in each. Check 90=`2×9×5`; 150=`2×3×25`. Discuss why a leftover 3 versus 5 matters.
4. **Choice · 6 min.** D18-A/B/C: compare original cards, factor-leaf match or critique a false reduction. Each keeps both totals and explains the division by 30.
5. **Exit · 4 min.** Complete `90/150=__/__` in simplest form: **3/5**. Name 30 as a shared factor and check `150×3/5=90`.
6. **Note · 2 min.** Record whether cancellation paired equal prime factors, not arbitrary digits.

**Optional home/extension:** Factor 90 and 150 on paper. Extension: find another common factor and show it reaches the same 3/5 after further reduction.

### Day 19 · Square products and their positive roots

**Codes:** `AC9M7N01`, `AC9M7N02`. **Goal:** connect `144=12²`, `√144=12` and prime powers. **Prepare:** [square-root link mat](print/square-root-link.pdf) or tactile grid.

1. **Launch · 2 min.** A fictional 12-by-12 pixel panel has how many cells? **144**.
2. **Model · 5 min.** `12²=12×12=144`, so `√144=12`. Factor 12=`2²×3`; squaring gives `144=2⁴×3²`. Check `16×9=144`. The root asks for the nonnegative side length, not `±12` in this setting.
3. **Guide · 6 min.** `100=10²`; 10=`2×5`, so 100=`2²×5²`, and `√100=10`. Compare 81=`9²`, 9=`3²`, so 81=`3⁴` and root9.
4. **Choice · 6 min.** D19-A/B/C: square grid, exponent pairs or square-root error repair. Require both multiplication and root explanation.
5. **Exit · 4 min.** “What is `√81`, and why?” **9** because `9×9=81`; `3⁴=81` does not mean the root is 3.
6. **Note · 2 min.** Check side/area units and distinguish evaluating a power from taking a square root; recheck 64 after instruction if needed.

**Optional home/extension:** Sketch 10×10 and 12×12 arrays. Extension: explain why `2⁴×3²` can be grouped into two equal `2²×3` products.

### Day 20 · Fresh prime-power and square-root check

**Codes:** `AC9M7N01`, `AC9M7N02`. **Goal:** transfer factorisation and square-root reasoning to unseen 108, 196 and 75. **Prepare:** closed [learner check](STUDENT-CHECKS.md), staff key, blank factor tree/root mat.

1. **Launch · 2 min.** Remind learners to multiply prime leaves back and test a root by squaring it.
2. **Model · 5 min.** Revisit 60=`2²×3×5` and 100=`10²`; put those models away. Keep check values unseen.
3. **Guide · 6 min.** Factor 36 as `2²×3²`, check `6²=36`, then put away. Do not display fresh numbers.
4. **Independent check · 6 min.** Give unseen Items 1–2. A blank tree or large/tactile cards may be supplied; do not choose factor leaves.
5. **Check exit · 4 min.** Item 3 asks for error detection in a proposed prime product. Preserve first work; D20 routes are later.
6. **Note · 2 min.** Record prime leaves, exponent counts, inverse square check and error explanation separately. A missing access format means not yet observed.

**Optional home/extension:** D20 extras after the check use different values; do not coach this held-out sheet at home.

The [optional extra cards](DAILY-EXTRAS.md) add short practice, not an entire mathematics day. Original SubjectNest lessons © NeuroForgeIO Pty Ltd 2026, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). Exact ACARA rows and taught limits appear in the [crosswalk](CURRICULUM-CROSSWALK.md).