# Teacher worked reasoning and next teaching (public resource)

This teacher-targeted key is publicly accessible in the library. Use it after collecting the first answer; change the case locally if prior access matters. Mark each criterion `0` absent/incorrect, `1` developing/with mathematical prompt, or `2` secure in this sample with a valid explanation; keep the actual support record alongside it. The rubric is not a total-score gate or whole Year 9 attainment decision. A learner may show the law by speech, AAC, symbol cards, writing, typing or directing a scribe. Do not score independent print reading from an exact adult read-aloud. A calculator verifies arithmetic but does not choose the law or domain.

## Daily feedback map

| Day | Evidence to recognise | If missing, teach next |
| ---: | --- | --- |
| 11 | `2^3×2^4=2^7=128` model strings because independent slots concatenate; `8+16=24` counts two lists, not pairs. | Pair two small lists physically; ask whether each first item meets every second item before writing powers. |
| 12 | `3^5/3^2=3^3=27` groups of nine; equal non-zero powers yield `a^0=1`. | Write the numerator and denominator as repeated factors, cancel only equal non-zero factors and label the quotient unit. |
| 13 | `10^−2=1/100=0.01`; 1 m model strip scaled gives 0.01 m or 1 cm. `(-2)^4=16`, `−2^4=−16`. | Convert a quotient to a fraction before using the negative-exponent notation; place sign inside/outside bracket cards. |
| 14 | `(2^3)^2=2^6=64`, while `2^3×2^2=2^5=32`; the source's second choice set differs. | Expand the outer power as two complete `2^3` groups; then compare with one `2^2` group. |
| 15 | Fresh Check A below; do not reuse the archive values as a teaching example before collection. | Choose A1–A4 next move and recheck with a different set of bases. |
| 16 | `x^3×x^2=x^5`; `x^5/x^2=x^3` only for x≠0. x2 checks 32 and 8; x0 invalidates only the quotient here. | Put PRODUCT and QUOTIENT on separate cards, then substitute zero into the **original** denominator. |
| 17 | `u^2/u^5=u^−3=1/u^3`, u≠0; u2 gives 1/8 and u−2 gives −1/8. | Build a cancellation ladder, then use brackets in odd-power signed substitutions. |
| 18 | `2^n×2^(n+2)=2^(2n+2)`; n2 gives 64, n0 gives 4. Exponents add as algebraic expressions. | Write n+(n+2) above the factor strings and distribute the addition before simplifying. |
| 19 | `(2^r)^2=2^(2r)` models a `2^r`-by-`2^r` grid; r3 gives 64. A two-row design gives `2^(r+1)=16` at r3. | Match a small drawn grid to each expression before giving any general law. |
| 20 | Fresh Check B below; model string pairs, unitless ratio and real readership remain separate. | Select B1–B4 next move; test the original denominator at zero if domain is omitted. |

## Check A Day 15 worked key

**A1 product:** `4^3×4^2=4^(3+2)=4^5=1024` **paired model label strings**. Arithmetic: 64×16=1024. The sum 64+16=80 counts elements in the separate lists if no pairing is made; it is not the Cartesian-product count under the source's each-with-every condition.

**A2 quotient:** `5^4/5^2=5^(4−2)=5^2=25` **groups of 25 paper cells** (625 cells ÷25 per group=25 groups). The source specifies equal groups. Exponents subtract because the **same non-zero base** factors cancel; adding would give `5^6`, not the quotient.

**A3 reciprocal:** `10^−3=1/10^3=1/1000=0.001`, a **positive unitless scale factor**. A negative exponent asks for a reciprocal, not a negative value.

**Worked correction:** “Each of 64 first label strings can pair with 16 second strings, so this invented model has 1024 pairs. The 625 paper cells make 25 groups of 25 because the same-base quotient subtracts exponents. The drawing scale is positive one-thousandth of a unit; none of these numbers describes a real archive or reader.” Equivalent valid wording earns the same evidence.

| Criterion | 0 · absent/incorrect | 1 · developing | 2 · secure in this sample | Immediate next move for 0/1 |
| --- | --- | --- | --- | --- |
| A1 · paired product | Adds 64+16 or adds bases. | Gives 1024 without pairing/same-base reason. | Shows `4^5=1024`, 64×16 and pair unit under each-with-every condition. | Build 2-by-3 smaller pair grid, then show each first item has all second choices. |
| A2 · grouped quotient | Adds exponents or calls 25 cells rather than groups. | Gives 25 but omits cancellation or unit. | Shows `5^(4−2)=25` groups, 625/25 check and non-zero common base. | Cancel two factors, label cells per group and number of groups distinctly. |
| A3 · negative exponent | Calls factor negative or uses `−1000`. | Gives fraction/decimal but cannot explain sign or reciprocal. | Shows `1/1000=0.001` and explains reciprocal of positive non-zero base. | Place 1 over 10³ and compare with 10²/10⁵. |
| A4 · editor correction | Accepts false source claims without a reason. | Names one wrong claim but leaves model/reality or operation unclear. | Corrects at least one false law with a source condition and keeps every conclusion fictional. | Order operation → same-base law → model unit → real-world limit. |

**Next teaching:** A1 absent → pairings/product meaning; A2 absent → division units and cancellation; A3 absent → reciprocal and sign; A4 partial with sound calculations → interpretation and source scope. Do not turn the eight possible rubric points into a Year 9 grade.

## Check B Day 20 worked key

**B1 variable product:** `p^(m+1)×p^(2m)=p^((m+1)+2m)=p^(3m+1)`. This uses equal base p and independent stage pairs. In the count context p is a positive whole and m a non-negative whole; the exponent expression is grouped.

**B2 substitution:** At p2, m2, first `2^(2+1)=2^3=8` strings; second `2^(2×2)=2^4=16`. Each-with-every gives 8×16=128 **model string pairs**. Simplified expression `2^(3×2+1)=2^7=128` agrees. This checks the sample but does not alone prove the identity.

**B3 reciprocal and zero:** `q^2/q^5=q^(2−5)=q^−3=1/q^3` for q≠0. At q2, `1/8`; at q0, original `0^2/0^5=0/0`, undefined. For negative q, the expression is still defined as long as q≠0; do not infer it is always positive.

**B4 claim:** The laws at q0 are not valid here because the quotient denominator is zero. A number of possible strings is not a count of people who use them; the source has no observed readers.

**Worked correction:** “With p positive and m a non-negative whole number, the two independent stages have `p^(3m+1)` possible pairs; the sample p2,m2 gives 128. The separate ratio is `1/q^3` only for q≠0, because the original quotient divides by `q^5`. Neither expression proves anyone used this fictional game.”

| Criterion | 0 · absent/incorrect | 1 · developing | 2 · secure in this sample | Immediate next move for 0/1 |
| --- | --- | --- | --- | --- |
| B1 · algebraic exponent | Adds or multiplies bases, or drops m+1 bracket. | Gets `3m+1` with no same-base/stage explanation. | Shows `(m+1)+2m=3m+1`, explains pair product and source domains. | Keep entire exponent phrases on separate cards; join with a visible plus. |
| B2 · value and unit | Incorrect 128 or calls it people. | Gets 128 only from one expression, not a check. | Gets 8×16=128 and `2^7=128` **model pairs**, not observed users. | Evaluate first and second stages separately, then multiply with pair unit. |
| B3 · quotient/domain | Says q0 valid or keeps a negative exponent without explanation. | Gives `1/q³` but omits q≠0 or q2 check. | Derives `q^−3=1/q³`, q≠0, q2=1/8, and denominator-zero reason. | Substitute zero into **original** q⁵ denominator, then use cancellation at q2. |
| B4 · claim boundary | Accepts universal law at zero or readership guarantee. | Rejects one claim without specific reason. | Corrects both zero-domain and unobserved-reader leaps using source wording. | Separate algebra DOMAIN and human OUTCOME headings, then ask what was measured. |

**Next teaching:** B1 fragile → combine algebraic exponents with brackets; B2 fragile → evaluate two stages and unit; B3 fragile → zero denominator/reciprocal; B4 fragile with sound algebra → distinguish possible strings from actual people. Preserve supported response modes while fading only mathematical cues.

## Optional quick key

| Day | Expected boundary |
| ---: | --- |
| 11 | `4^2×4^1=4³=64`; `2²×2⁵=128`, while `2²+2⁵=36`. |
| 12 | `7⁴/7²=7²=49`; `9³/9³=1` for non-zero base 9. |
| 13 | `10⁻¹=0.1`; `3⁻²=1/9`, not negative. |
| 14 | `(4²)²=4⁴=256`; `(5¹)³=5³=125`, `5¹×5³=5⁴=625`. |
| 15 | Optional tasks return to Cards A/B only **after** Check A collection. |
| 16 | `y²×y⁴=y⁶`; `z⁴/z²=z²` only z≠0. |
| 17 | `w/w⁴=w⁻³`, at w−1 result −1; `(1/2)⁻¹=2` for non-zero base. |
| 18 | `3^t×3^(t+1)=3^(2t+1)`; at t1, 27. At k1, `2^(k+1)×2^(k+1)=16` versus `2^(k+2)=8`. |
| 19 | `(4^j)^2=4^(2j)` and j1 yields 16; two columns of 3^s at s1 make 6, not 9. |
| 20 | Optional tasks return to Cards F/H **after** Check B collection. |

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