# Year 9 mathematics · ten 25-minute lessons · Days 11–20

Use the original [model cards](MODEL-CARDS.md), [clean learner routes](LEARNER.md), [optional extras and no-purchase home ideas](EXTRAS-AND-HOME.md), [A4/text/tactile aids](print/TEXT-ALTERNATIVES.md) and public teacher [worked key](TEACHER-KEY.md). Give [fresh Check A/B](ASSESSMENT.md) as clean prompts on Days 15/20; the key is also public, so adapt a case locally if prior access matters. This continues the opening fortnight's attention to coordinates, rates, domains and model limits; Week 3 studies **numerical integer exponents**, then Week 4 **variable bases and exponents**. Every lesson is 2 + 5 + 6 + 6 + 4 + 2 = **25 minutes**. Allow a calculator for arithmetic access while asking for the structure, condition and unit. Read-aloud is listening-supported access, **not independent print reading**. Count speech, AAC, tactile cards, typing and directed scribing as the learner's mathematics when their choices remain theirs; record any mathematical hint. All contexts and numbers are classroom fiction.

## Week 3 Day 11 Product of powers counts combinations

**Code:** `AC9M9A01`. **Goal:** derive same-base product law from repeated independent choices and reject an invalid sum shortcut. **Prepare:** [Card A](MODEL-CARDS.md#card-a-nested-message-choices), [law and non-law mat](print/law-boundary.pdf), two sets of yes/no paper tokens.

1. **Retrieve · 2 min.** Ask what `2^3` counts and what the exponent 3 tells us. State base 2 and three independent binary slots, not “2×3”.
2. **Model · 5 min.** Arrange a three-slot and four-slot string. Joining them makes seven slots, so `2^3×2^4=2^(3+4)=128`. Multiply 8×16 as a check. Name model **strings**, not real player behaviour.
3. **Guided · 6 min.** Try `3^2×3^3=3^5=243` by writing two and three factors of 3. Compare `3^2+3^3=9+27=36`; the product law does not cross a plus sign.
4. **Choose · 6 min.** Learners take [11 A/B/C](LEARNER.md#day-11): token sequence, factor explanation or wrong-post repair. Each must state why the bases match and give one check.
5. **Probe · 4 min.** Ask whether `2^3×3^4` permits simply adding exponents. It has different bases, so evaluate or reorganise only with a justified property; no same-base shortcut.
6. **Exit · 2 min.** “The product works because ___; the plus version ___.” If learner adds 8+16 to count pairs, make a two-row pairing grid before new values.

## Week 3 Day 12 Quotient, zero exponent and the unit of the answer

**Code:** `AC9M9A01`. **Goal:** derive same-base quotient and zero-exponent rules from equal-sized groups. **Prepare:** [Card B](MODEL-CARDS.md#card-b-grouping-a-paper-cell-array), [quotient ladder](print/quotient-domain.pdf), paper cells/group labels.

1. **Retrieve · 2 min.** Expand `3^2` and `3^5` as repeated factors. Ask what a division `243÷9` counts in Card B.
2. **Model · 5 min.** Cancel two non-zero factors of 3 from `3^5/3^2` to obtain `3^3=27` groups of 9 cells. State the model count and quotient unit.
3. **Guided · 6 min.** `5^3/5^3=1` exactly, so `5^(3−3)=5^0=1`. This explanation requires non-zero 5; `0^0` is outside this rule. Then `2^6/2^2=2^4=16` with a factor check.
4. **Choose · 6 min.** Learners use [12 A/B/C](LEARNER.md#day-12): physical grouping, error analysis or verbal ratio. Require numerator/denominator and why subtraction applies.
5. **Probe · 4 min.** Ask whether `3^5/2^2` permits exponent subtraction. Different bases: no. A calculator may give a number but not justify a false rule.
6. **Exit · 2 min.** “For a non-zero same base, `a^m/a^n=___`; if exponents match, ___.” Attend to group units if the rule is correct but meaning is not.

## Week 3 Day 13 A negative exponent is a reciprocal

**Code:** `AC9M9A01`. **Goal:** use quotient reasoning to rewrite a negative integer exponent as a positive reciprocal and maintain units. **Prepare:** [Card C](MODEL-CARDS.md#card-c-a-small-paper-scale), [quotient ladder](print/quotient-domain.pdf), one paper metre strip drawing.

1. **Retrieve · 2 min.** Use `10^2/10^4=10^(2−4)` and ask what ordinary fraction this quotient represents.
2. **Model · 5 min.** Cancel factors: `100/10000=1/100=10^−2`. Scale a **drawn** 1 m strip by this dimensionless factor to `0.01 m = 1 cm`. Mark centimetre as a unit, not part of the exponent law itself.
3. **Guided · 6 min.** `2^−3=1/2^3=1/8`; `5^−1=1/5`. Ask why a negative exponent does not make the result negative. For a negative base, `(-2)^−3=−1/8`; parentheses make the base negative.
4. **Choose · 6 min.** Take [13 A/B/C](LEARNER.md#day-13): reciprocal tiles, scale annotation or critique `10^−2=−100`. Each includes the non-zero-base condition.
5. **Probe · 4 min.** Compare `(-2)^4=16` with `−2^4=−16`. Show the brackets, not just a calculator key sequence. Do not apply `0^−1`.
6. **Exit · 2 min.** Rewrite `4^−2` as a fraction and explain the base condition: `1/16`, base non-zero.

## Week 3 Day 14 A power of a power is repeated groups

**Code:** `AC9M9A01`. **Goal:** distinguish `(a^m)^n=a^(mn)` from a product of two powers, using brackets and a model. **Prepare:** [Card D](MODEL-CARDS.md#card-d-cue-combinations), [law boundary mat](print/law-boundary.pdf).

1. **Retrieve · 2 min.** Name how many choices `2^3` represents in the fictional stage notebook.
2. **Model · 5 min.** Two independent eight-choice cards give `(2^3)^2=2^3×2^3=2^6=64`. Multiplying exponents is a shorthand for repeating the **whole** power twice.
3. **Guided · 6 min.** Compare `2^3×2^2=2^5=32`, a different second stage. Then `(3^2)^3=3^6=729`; expand as three groups of two factors to check.
4. **Choose · 6 min.** Take [14 A/B/C](LEARNER.md#day-14): bracket tree, contrasting examples or caption editor. Each states exactly which expression matches which fictional cue structure.
5. **Probe · 4 min.** Contrast `(2^3)^2` with `2^(3+2)`. Ask where the repeated group lies in each. Keep `(-2)^4` brackets distinct from `−2^4`.
6. **Exit · 2 min.** Write one sentence explaining why a power of a power multiplies exponents; show a small expanded example.

## Week 3 Day 15 Fresh numerical exponent check

**Code:** `AC9M9A01`. **Goal:** independently choose exponent laws for an new numerical source and explain one invalid claim. **Prepare:** fresh [Check A](ASSESSMENT.md#check-a-day-15-fresh-number-source) and public teacher [worked key](TEACHER-KEY.md#check-a-day-15-worked-key). No practice with its values.

1. **Frame · 2 min.** State that the new card is a fresh source. Learners can use a blank law mat, calculator and their usual access method, with mathematical hints noted.
2. **Read · 5 min.** Give the source exactly; an adult read-aloud is logged as such. Do not identify the useful law.
3. **First work · 6 min.** Learners mark bases, operation signs and any denominator before simplifying. Preserve first response.
4. **Respond · 6 min.** Choose [15 A/B/C](LEARNER.md#day-15) for the same product, quotient, reciprocal and claim-boundary thinking in the learner's own words.
5. **Audit · 4 min.** Ask students to compare one result with expanded factors or substitution without telling them which result to change. Keep revisions visible.
6. **Collect · 2 min.** Use criterion next moves; a small check cannot decide Year 9 attainment.

## Week 4 Day 16 Variable bases and independent slots

**Code:** `AC9M9A01`. **Goal:** extend product and quotient laws to a variable base and distinguish the product's domain from the quotient's. **Prepare:** [Card E](MODEL-CARDS.md#card-e-variable-choices-in-independent-slots), [variable-domain aid](print/variable-power-map.pdf).

1. **Retrieve · 2 min.** Replace each abstract slot with `x` choices. Ask what three independent slots count when x is a positive whole number.
2. **Model · 5 min.** Join three and two slots: `x^3×x^2=x^5`. At x2, 8×4=32=2^5. The product identity also makes sense at x0 with these positive exponents, though the count context has no strings.
3. **Guided · 6 min.** Divide `x^5/x^2=x^3` by cancelling non-zero x factors. At x2, 32/4=8. For x0 the original quotient is 0/0, undefined; write `x≠0` before cancellation.
4. **Choose · 6 min.** Use [16 A/B/C](LEARNER.md#day-16): slot diagram, algebra audit or domain comparison. Each gives a substitution check and explicit quotient restriction.
5. **Probe · 4 min.** Ask if `x^3+y^2` may combine exponents. No: different bases and addition. Keep “cannot use this rule” separate from “cannot evaluate given values”.
6. **Exit · 2 min.** State one valid product identity and one domain condition for a variable quotient.

## Week 4 Day 17 Negative variable exponents and signs

**Code:** `AC9M9A01`. **Goal:** rewrite a negative variable exponent as a reciprocal and test positive/negative non-zero substitutions. **Prepare:** [Card F](MODEL-CARDS.md#card-f-a-non-zero-scale-ratio), [quotient ladder](print/quotient-domain.pdf).

1. **Retrieve · 2 min.** Ask whether `u^2/u^5` divides by u when u0. Name the restriction first.
2. **Model · 5 min.** For u≠0, cancel: `u^2/u^5=u^(−3)=1/u^3`. At u2, result 1/8. Keep it a **unitless ratio**, not a count or length.
3. **Guided · 6 min.** At u−2, `u^3=−8`, so reciprocal `−1/8`. At u0, original denominator `0^5=0`, so no value. Contrast `(-u)^2` and `−u^2` at u2.
4. **Choose · 6 min.** Take [17 A/B/C](LEARNER.md#day-17): cancellation ladder, error critique or signed substitution. Each includes u≠0 and fraction/negative-exponent forms.
5. **Probe · 4 min.** Challenge “negative exponent means negative result” with u2 and u−2. The sign comes from the base and parity, not the negative exponent by itself.
6. **Exit · 2 min.** Rewrite `v^−2` as `1/v^2` with `v≠0`; say what fails at v0.

## Week 4 Day 18 A variable appears inside the exponent

**Code:** `AC9M9A01`. **Goal:** add algebraic exponents for equal bases and test a whole-number context. **Prepare:** [Card G](MODEL-CARDS.md#card-g-a-variable-exponent-in-a-game-rule), [variable-power map](print/variable-power-map.pdf).

1. **Retrieve · 2 min.** Read `2^(n+2)` as a power with **n+2** in the exponent; do not treat it as `2^n+2`.
2. **Model · 5 min.** Join independent stages: `2^n×2^(n+2)=2^(n+n+2)=2^(2n+2)`. In Card G, n is a non-negative whole stage number.
3. **Guided · 6 min.** At n2, `2^2×2^4=4×16=64=2^6`; at n0, `1×4=4=2^2`. Both checks match the formula but do not replace the exponent-law argument.
4. **Choose · 6 min.** Use [18 A/B/C](LEARNER.md#day-18): stage table, algebra-line repair or spoken/tactile path account. Each writes the bracketed exponent and one domain statement.
5. **Probe · 4 min.** Compare `2^(2n+2)` with `2^(n^2+2)` at n3: exponents 8 and 11, not equal. Ask where the extra n×n came from; none is in the product law.
6. **Exit · 2 min.** Simplify `3^m×3^(m+1)=3^(2m+1)` and state that m is a whole number in a slot-count model.

## Week 4 Day 19 Choose a law for the design, not the other way round

**Code:** `AC9M9A01`. **Goal:** select the correct exponent structure for an original poster grid and explain a nearby non-example. **Prepare:** [Card H](MODEL-CARDS.md#card-h-poster-icon-grid), [law-boundary](print/law-boundary.pdf) and [variable-power](print/variable-power-map.pdf) aids.

1. **Retrieve · 2 min.** Ask whether the design has two rows or `2^r` rows. The source, not the symbol alone, decides.
2. **Model · 5 min.** A `2^r`-by-`2^r` grid contains `(2^r)^2=2^(2r)` icons. At r3, 8×8=64. A **different** two-row design would contain `2×2^r=2^(r+1)`; label the changed condition.
3. **Guided · 6 min.** Check r2: first design 4×4=16, second 2×4=8. Discuss why a model icon count cannot guarantee legible printing, physical space or reader access.
4. **Choose · 6 min.** Take [19 A/B/C](LEARNER.md#day-19): compare grid sketches, create a different exponent model, or diagnose a misleading caption. Each names the count unit, r domain and what is unmodelled.
5. **Audit · 4 min.** Peer/teacher asks whether the learner's drawing represents the same rule they wrote. If not, revise either the words or expression, keeping first work.
6. **Exit · 2 min.** “I chose ___ because the source repeats ___; I would not claim ___ from the count alone.”

## Week 4 Day 20 Fresh variable-exponent transfer check

**Code:** `AC9M9A01`. **Goal:** use the integer exponent laws in an new variable source with honest domains and one model limit. **Prepare:** fresh [Check B](ASSESSMENT.md#check-b-day-20-fresh-variable-source) and separate [worked key](TEACHER-KEY.md#check-b-day-20-worked-key).

1. **Frame · 2 min.** State that this is a fresh card. Blank aid, calculator or usual accessible representation may be used; keep mathematical prompting visible.
2. **Read · 5 min.** Give new source exactly. Record exact adult reading versus independent print. Do not model its values first.
3. **First work · 6 min.** Learners mark base, exponent expression, division and any zero-base exclusion. Keep unprompted work.
4. **Respond · 6 min.** Choose [20 A/B/C](LEARNER.md#day-20) for the same algebra, substitution, reciprocal and source-reach criteria; preserve their wording.
5. **Audit · 4 min.** Ask for one substitution or expanded-factor check, then a second line distinguishing model count from a real-world promise. Record any change.
6. **Collect · 2 min.** Use the criterion-specific staff key for subsequent teaching, not a total score as Year 9 certification.

**Original resource rights:** © NeuroForgeIO Pty Ltd 2026, SubjectNest, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). Credit author, source, licence and changes. ACARA wording is separately attributed in the crosswalk.