# Year 6 mathematics · Days 11–20 · ten 25-minute lessons

**Daily rhythm:** launch 2 + model 5 + guided 6 + one learner route 6 + exit 4 + evidence note 2 = **25 minutes**. On Days 15 and 20, the six-minute route and four-minute exit become a ten-minute fresh check; the three routes on those days are **later practice**. Use [clean learner prompts](LEARNER.md), [three daily routes](DAILY-CHOICES.md), original [A4 aids and their full alternatives](print/TEXT-ALTERNATIVES.md), and the [staff key](teacher/ANSWER-AND-NEXT.md). Record whether the learner reasoned independently, used a reader/scribe or received a mathematical hint. Do not mistake an access route for weaker reasoning or a single check for an achievement-standard verdict.

## Week 3 · Properties of prime, composite and square numbers

**Teacher convention:** a *prime* is a whole number greater than 1 with exactly two positive factors; a *composite* is a whole number greater than 1 with more than two positive factors; a *square number* is `n × n` for a whole number `n` (this fortnight uses positive `n`). A square number greater than 1 is composite. **1 is square (`1 × 1`) but neither prime nor composite.** The words “prime/composite” classify differently from “square”: 4 is both composite and square. Material and safety choices are in [MATERIALS](MATERIALS.md).

### Day 11 · Classify by factors, not appearance

**Code:** `AC9M6N02`. **Goal:** use a factor-array explanation to classify 1, 2, 12 and 16. **Prepare:** [blank factor-array lab](print/factor-array-lab.pdf) or text/tactile route, 16 counters/cards.

1. **Launch · 2 min.** Ask whether a number can be both square and composite; collect a tentative answer without rating speed.
2. **Model · 5 min.** Arrange 12 as `1 × 12`, `2 × 6`, `3 × 4`; show factor pairs. Twelve has more than two positive factors, so composite. Arrange 16 as `4 × 4`; it is square **and** composite (`2 × 8` also works). Say why 1 has just one positive factor and 2 has exactly two.
3. **Guided · 6 min.** Learners use tiles, drawn cells or spoken factor pairs for 1 and 2, then check 16. Name the rule **greater than 1** for both prime and composite. Do not call “odd” a prime test.
4. **Choice · 6 min.** One D11-A/B/C [route](DAILY-CHOICES.md): each requires a correct classification and factor evidence.
5. **Exit · 4 min.** “Is 16 prime because its square array looks special?” **No**, since 16 has factors 1, 2, 4, 8, 16. “Is 1 prime?” **No**, it has one positive factor.
6. **Note · 2 min.** Save factor language and a diagram/spoken explanation. If labels are memorised without proof, reteach with a new array.

**Home/extension:** Make 12 with bottle-cap substitutes or sketch boxes; no purchase. Extension: explain why every positive square above 1 has at least factors `1, n, n²` (for `n>1`, these are distinct).

### Day 12 · A complete factor check

**Code:** `AC9M6N02`. **Goal:** find factor pairs without skipping one and justify why 23 is prime. **Prepare:** scratch grid, D12 routes.

1. **Launch · 2 min.** Show 18 and ask for two different rectangular row plans.
2. **Model · 5 min.** List 18 as `1 × 18`, `2 × 9`, `3 × 6`; stop when the next trial would repeat a pair. For 23, test possible small divisors 2, 3 and 4; none divide evenly, and `5 × 5 > 23`, so an unseen factor pair cannot begin at 5 or more. Therefore only `1 × 23` and 23 is prime. “Not obvious” alone is no proof.
3. **Guided · 6 min.** Check 20's factor pairs `1 × 20`, `2 × 10`, `4 × 5`; compare 21's `1 × 21`, `3 × 7`. Learners explain how pairs link to arrays, not only a yes/no label.
4. **Choice · 6 min.** D12-A/B/C, using seats, archive trays or a number detective game; check the full pair list or a justified prime decision.
5. **Exit · 4 min.** “Why can 23 have no pair `5 × something whole`?” Because `5 × 5` already exceeds 23 and the smaller member of any factor pair must be at most the square-root boundary. Accept a concrete pair-table explanation without requiring square-root notation.
6. **Note · 2 min.** Record whether the learner tried all needed small divisors and knew when to stop; recheck with a fresh two-digit number.

**Home/extension:** Draw rectangles for 18/20. Extension: compare how `2 × 11 = 22` and `3 × 8 = 24` flank prime 23 without treating neighbours as proof.

### Day 13 · Squares belong in two conversations

**Code:** `AC9M6N02`. **Goal:** locate square numbers in a factor grid and state when they are composite. **Prepare:** [square-and-factor strip](print/square-factor-strip.pdf) or tactile cards.

1. **Launch · 2 min.** Show `1 × 1`, `3 × 3`, `7 × 7`; learners predict their totals.
2. **Model · 5 min.** Fill `1, 4, 9, 16, 25, 36, 49` as squares of 1–7. Explain 49 as `7 × 7` **and** composite since factors include 1, 7, 49. State the exception: 1 is square but neither prime nor composite.
3. **Guided · 6 min.** Investigate 36 with `6 × 6` and another array `4 × 9`; use two routes to justify both labels. Sort 9 and 17; 9 is square/composite, 17 prime/not square. A learner may speak equations instead of moving small counters.
4. **Choice · 6 min.** D13-A/B/C: square mosaic, factor-card sort or a careful reply to “all squares are prime”.
5. **Exit · 4 min.** “Can a positive square greater than 1 be prime?” **No**: `n × n` gives a factor `n` between 1 and `n²`. Ask for 25 or 49 as an example.
6. **Note · 2 min.** Keep evidence for square property and prime/composite property separately; reteach the exception with 1 if necessary.

**Home/extension:** Draw 4×4 and 5×5 on grid paper. Extension: compare consecutive squares 25 and 36 using visible border growth; no general algebra rule is required here.

### Day 14 · Use properties to choose a plan

**Code:** `AC9M6N02`. **Goal:** use factor/square properties in a practical decision and simplify a product. **Prepare:** fictional event layout cards, D14 choices.

1. **Launch · 2 min.** A fictional gallery has 28 display cards. Can it have more than one equal row and column? **Yes**, e.g. `4 × 7`.
2. **Model · 5 min.** Compare 29 cards: no pair of whole rows/columns with both sides >1 (`2,3,4,5` do not divide; next square `6²>29`), so a complete 29-card rectangle is only `1 × 29`. For 36 cards, `6 × 6` makes a square and `4 × 9` another rectangle. Show mental `12 × 15 = (3 × 4) × (3 × 5) = 9 × 20 = 180`; re-grouping uses factor properties without changing the product.
3. **Guided · 6 min.** Give 24 rehearsal seats: list `2 × 12`, `3 × 8`, `4 × 6`; decide which fits a six-seat-wide area. It is `4 × 6`, with no assumptions about actual room access or capacity.
4. **Choice · 6 min.** D14-A/B/C: gallery, game-board or calculation route. Require a factor statement **and** a reason tied to the context.
5. **Exit · 4 min.** “For 36 items, is `6 × 6` the only rectangle?” **No**; name `4 × 9` or `3 × 12`. “What is `12 × 15`?” **180**, by a checked regrouping.
6. **Note · 2 min.** Check if the learner confused “a square exists” with “only one array exists”; use 16 as a new counterexample.

**Home/extension:** Design a fictional 24-cell board on paper. Extension: explain which factors make 36 quick to multiply without changing a product.

### Day 15 · Fresh number-properties check

**Code:** `AC9M6N02`. **Goal:** transfer classification and factor reasoning to unseen 27, 37, 64, 31 and 33. **Prepare:** [fresh learner check](STUDENT-CHECKS.md), separate [staff key](teacher/ANSWER-AND-NEXT.md). Do not show its numbers in practice first.

1. **Launch · 2 min.** Tell learners a new set helps decide the next teaching move; access tools and a quiet response are available.
2. **Model · 5 min.** Revisit **different** numbers: 9 is `3 × 3` and composite; 17 has no 2, 3 or 4 factor and is prime. Put examples away.
3. **Guided · 6 min.** Use 20: `1 × 20`, `2 × 10`, `4 × 5`. Ask why it is composite but not square. Keep fresh-check values unseen.
4. **Independent check · 6 min.** Give unseen Day 15 Items 1–2. Read wording neutrally if needed; do not offer a factor or label.
5. **Check exit · 4 min.** Items 3–4 ask for a square/composite overlap and a layout decision. Save first answers before feedback; later D15 routes are separate practice.
6. **Note · 2 min.** Score definitions, factor evidence, square overlap and contextual choice separately. If access/time interrupted, mark not yet observed rather than incorrect.

**Home/extension:** After the first check response, optional routes can go home. Do not turn the held-out sheet into a coached homework task.

## Week 4 · Equivalent common fractions on the same whole and line

Use a **fixed unit whole** and equal-length sections. On a 0–1 line divided into 12 equal gaps, `1/4=3/12`, `1/3=4/12`, `1/2=6/12`, `2/3=8/12`, `3/4=9/12`. The same mark can have two fraction names. The [12-gap and 24-gap lines](print/fraction-lines.pdf) and [equal-whole bars](print/equal-whole-bars.pdf) are blank learner aids; the [full text/tactile route](print/TEXT-ALTERNATIVES.md) gives scales without pre-filling answers. A 0–2 printed line has a different physical scale than a same-length 0–1 printed line; compare *fraction values*, not centimetres across those two diagrams.

### Day 16 · Fix the whole before comparing

**Code:** `AC9M6N03`. **Goal:** locate `1/4`, `1/3`, `1/2` in order on one 0–1 line. **Prepare:** fraction line and equal-whole bars or 12 equal tactile strips.

1. **Launch · 2 min.** Ask why “one half is bigger than one third” needs the same-size whole.
2. **Model · 5 min.** Mark 0 and 1, then 12 equal spaces. Place `1/4` at gap 3, `1/3` at gap 4, `1/2` at gap 6. Explain `3/12 < 4/12 < 6/12`; point to the same endpoint 1 for all.
3. **Guided · 6 min.** Learners use bars split into 2, 3, 4 or 12 equal parts and map `1/2=6/12`, `1/3=4/12`. Ask whether gap 6 on a **different** 6-gap 0–1 line has the same value as gap 6 on this 12-gap line. **No:** `6/6=1`, while `6/12=1/2`; the scale matters.
4. **Choice · 6 min.** D16-A/B/C: line, recipe-share cards or a musical-cycle fraction route; each uses one whole and justifies order.
5. **Exit · 4 min.** On the same whole, order `1/2`, `1/4`, `1/3`: **`1/4 < 1/3 < 1/2`**. Explain by twelfths or equal bars, not denominator size alone.
6. **Note · 2 min.** Record whether endpoints and equal spacing were checked. If not, restore a shared whole before rechecking.

**Home/extension:** Fold a paper strip into 2 then 4 equal parts, or draw 12 gaps; no food or instrument needed. Extension: explain why differently sized physical wholes make direct piece comparisons unsafe.

### Day 17 · Equivalent names, one position

**Code:** `AC9M6N03`. **Goal:** show equivalent fractions and compare `2/3` and `3/4` using one 12-gap line. **Prepare:** bars/line or spoken gap sequence.

1. **Launch · 2 min.** Ask whether `2/4` and `1/2` need two marks on the same 0–1 line.
2. **Model · 5 min.** Place `2/4=1/2=6/12` at one mark. Place `2/3=8/12` and `3/4=9/12`; `3/4 > 2/3` by **1/12** of the same whole. Say numerator and denominator both scale by the same factor when making equivalent names.
3. **Guided · 6 min.** Match `1/3=4/12`, `2/4=6/12`, `3/4=9/12`; have learners correct a claim that `2/3 > 3/4` merely because 3 is greater than 4 in a numerator/denominator comparison.
4. **Choice · 6 min.** D17-A/B/C: line explanation, archive-space diagram or original game-state explanation. Require equivalent names and same whole.
5. **Exit · 4 min.** “Which is larger, `2/3` or `3/4`, and by how much?” **`3/4` by `1/12`**. Ask where both sit between 0 and 1.
6. **Note · 2 min.** Check whether learner can justify order, not only convert by rote. Recheck with `1/3` versus `1/2` if needed.

**Home/extension:** Use one folded strip as the whole. Extension: find `3/6` and `6/12` on the same half mark, with equal partitions shown.

### Day 18 · Fractions beyond one on the same scale

**Code:** `AC9M6N03`. **Goal:** represent and compare `5/4` and `3/2` on a 0–2 line. **Prepare:** [24-gap line](print/fraction-lines.pdf) or tactile equivalents, two equal unit strips.

1. **Launch · 2 min.** Ask whether a fraction greater than 1 can still be marked on a number line. **Yes**, if the line extends beyond 1.
2. **Model · 5 min.** On 0–2 with 12 equal gaps in **each unit**, mark `5/4 = 1 + 1/4 = 15/12` at gap 15, and `3/2 = 1 + 1/2 = 18/12` at gap 18. Thus `5/4 < 3/2` by `3/12=1/4`.
3. **Guided · 6 min.** Locate `1 1/2` at the same gap as `3/2` and `6/4`. Locate `1 1/4` with `5/4`. Ask why a second whole must match the first whole's length.
4. **Choice · 6 min.** D18-A/B/C: extended line, 12-beat loop over two cycles or art-tile strip. Show an equivalence and a reason for order.
5. **Exit · 4 min.** “Is `5/4` to the left or right of `3/2`?” **Left**. “How far?” **`1/4` of the unit whole**.
6. **Note · 2 min.** Record if learner labelled 1 at gap 12 and 2 at gap 24; recheck scale before fraction work.

**Home/extension:** Draw two matching strips and split each into four; extension: explain why `6/4=3/2` is one mark, not two.

### Day 19 · Justify an order and repair a false rule

**Code:** `AC9M6N03`. **Goal:** use equivalent names to order common fractions and diagnose denominator-only reasoning. **Prepare:** 12-gap line/bars, D19 routes.

1. **Launch · 2 min.** Display false claim: “`1/3 > 1/2` because 3 > 2.” Ask which whole/mark would disprove it.
2. **Model · 5 min.** On the same 0–1 whole, `1/3=4/12` and `1/2=6/12`, so `1/3 < 1/2`. A larger denominator of a **unit fraction** means smaller equal parts when the whole stays fixed. Do not turn this into an unsafe rule for every pair of fractions.
3. **Guided · 6 min.** Order `1/4, 1/2, 2/3` as `3/12,6/12,8/12`. Learners compare another same-whole pair `3/4` and `1/2` using marks 9 and 6, then say why a different whole would change physical sizes.
4. **Choice · 6 min.** D19-A/B/C: evidence to a fictional exhibit maker, an error-repair dialogue or a tactile same-whole sequence. State an exact comparison and explain it.
5. **Exit · 4 min.** A peer places `2/4` to the **right** of `1/2` on one line. Repair: both equal `6/12` and occupy **the same mark**.
6. **Note · 2 min.** Distinguish a correct order with reasoning from a memorised slogan. Recheck a new pair without the model in view.

**Home/extension:** Draw a new line and give an adult a deliberate error to repair. Extension: show `3/6`, `2/4` and `1/2` as one point with same-size whole.

### Day 20 · Fresh fraction-line check

**Code:** `AC9M6N03`. **Goal:** transfer equivalent-name, order and same-whole reasoning to unseen `5/3` and `7/4` plus a new 24-tile context. **Prepare:** closed [learner check](STUDENT-CHECKS.md), blank line/strips, separate [staff key](teacher/ANSWER-AND-NEXT.md).

1. **Launch · 2 min.** Remind learners to mark 0, 1 and 2 before deciding an order; response mode is open.
2. **Model · 5 min.** Revisit different numbers: `5/4=15/12`, `3/2=18/12`; put away. Do not show the check's `5/3` or `7/4`.
3. **Guided · 6 min.** Use `1/4=3/12` and `1/2=6/12` on one fixed 0–1 line. Ask why six gaps on a 12-gap line represent half. Put it away.
4. **Independent check · 6 min.** Give unseen Day 20 Items 1–2. Adult can read wording neutrally or provide a tactile line without placing answers.
5. **Check exit · 4 min.** Item 3 asks for a same-whole comparison and a false-rule repair. Preserve first response and route; D20 choices are **later practice**.
6. **Note · 2 min.** Mark endpoints/equal scale, equivalent names, order and justification separately. If learner had no usable access route, mark not yet observed and arrange a valid later check.

**Home/extension:** Fresh check stays in class. Afterward, optional D20 routes use different numbers and a learner-selected everyday context.

**Later in the day:** The ten [optional extra cards](DAILY-EXTRAS.md) give short practice or transfer, not a full mathematics timetable. No learner needs a home device, purchase or personal finance disclosure. The exact curriculum descriptions and taught limits are in the [crosswalk](CURRICULUM-CROSSWALK.md). Original SubjectNest lessons © NeuroForgeIO Pty Ltd 2026, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/).