# Public teacher-targeted worked key and next moves

This key is reachable by anyone with the pack URL. The two checks are fresh relative to practice, **not secure examinations**. Collect first responses before displaying a model. Accept equivalent exact forms only when the expression is genuinely equal; approximate values require `≈`. Spoken and tactile routes still need captured exact notation and justification.

## Card answers and teaching checks

| Card | Worked result and important reason |
|---|---|
| A | `sqrt(9)=3`, `sqrt(16)=4`; `sqrt(10)` is irrational and is a surd because 10 is not a perfect square. `3<sqrt(10)<4`. `sqrt(10)≈3.162`, never exact equality to rounded 3.16. |
| B | `72=36×2`, so `sqrt(72)=6sqrt(2)`, and `(6sqrt(2))²=36×2=72`. Since `64<72<81`, `8<sqrt(72)<9`. Square-root splitting does not apply to a sum. |
| C | `sqrt(50)=5sqrt(2)`, `sqrt(98)=7sqrt(2)`; second longer by `2sqrt(2)` units. |
| D | `sqrt(12)=2sqrt(3)` and `sqrt(27)=3sqrt(3)`, giving `5sqrt(3)` cm, about 8.66 cm. `sqrt(39)` lies between 6 and 7, so it cannot be the sum. `sqrt(12)+sqrt(8)=2sqrt(3)+2sqrt(2)` cannot combine further as like surds. |
| E | `sqrt(32)-sqrt(8)=4sqrt(2)-2sqrt(2)=2sqrt(2)` cm. `sqrt(8)+sqrt(18)=2sqrt(2)+3sqrt(2)=5sqrt(2)` cm. |
| F | `sqrt(6)sqrt(15)=sqrt(90)=3sqrt(10)` cm², approximately 9.49 cm². `(2sqrt(3))(3sqrt(6))=6sqrt(18)=18sqrt(2)`. |
| G | `7/sqrt(5)×sqrt(5)/sqrt(5)=7sqrt(5)/5`; top and bottom are multiplied by the same nonzero factor, so value is unchanged. `4/sqrt(2)=4sqrt(2)/2=2sqrt(2)`. |
| H | `sqrt(18)=3sqrt(2)`, `sqrt(8)=2sqrt(2)`, so the fraction is `5sqrt(2)/sqrt(2)=5`. Alternatively `sqrt(18)/sqrt(2)+sqrt(8)/sqrt(2)=sqrt(9)+sqrt(4)=3+2=5`. The invalid draft replaces a sum of roots by the root of a sum. `sqrt(26)/sqrt(2)=sqrt(13)≈3.61`, not 5. |
| I | Table for x=0,1,2,3,4 gives y=5,2,1,2,5. Turning point `(2,1)`, axis `x=2`, y-intercept `(0,5)`, no real x-intercepts because `(x-2)²+1≥1`. |
| J | Zeros `x=-1,3`; axis `x=1`; turning point `(1,-4)` from `(2)(-2)=-4`; y-intercept `(0,-3)`; opens up because x² coefficient is +1; y<0 for `-1<x<3`. |

## Check A · Day 5

`243=81×3` gives `sqrt(243)=9sqrt(3)` cm. `48=16×3` gives `sqrt(48)=4sqrt(3)` cm. Difference is `5sqrt(3)` cm, approximately `8.7` cm to one decimal (`5×1.73205…≈8.66025`). The draft's `sqrt(195)` is approximately 14.0 cm; it is not `5sqrt(3)`. More fundamentally, `sqrt(a)-sqrt(b)` cannot be turned into `sqrt(a-b)` for these nonnegative a,b. A valid exact comparison squares positive values: `(5sqrt(3))²=75`, while `(sqrt(195))²=195`.

**Respond:** if factor extraction fails, supply a square-number list and ask for `81×3` and `16×3`. If terms are combined as `sqrt(195)`, ask the learner to square both proposed positive values. If the calculation is correct but `=` is used for 8.7, practise exact versus approximate notation. If secure, use a fresh invented unlike-radical pair and ask why it cannot combine.

## Check B · Day 10

For `y=-(x-2)(x-8)`, zeros are `(2,0)` and `(8,0)`. The axis is their midpoint `x=5`. At x=5, `y=-[(3)(-3)]=9`, so the turning point is `(5,9)`. At x=0, `y=-[(-2)(-8)]=-16`, so y-intercept `(0,-16)`. The negative x² coefficient makes the parabola open downward. Between the zeros, e.g. x=5, the product `(x-2)(x-8)` is negative, so its negative is positive; the graph is above the x-axis for **`2<x<8`** and equals zero at the endpoints. `5/sqrt(11)×sqrt(11)/sqrt(11)=5sqrt(11)/11`; `sqrt(11)` is nonzero, so the multiplier is 1.

**Respond:** if the midpoint 5 is reported as the turning *height*, substitute x=5 to obtain 9. If the sign interval is reversed, test x=5 and x=0. If a learner rationalises only the numerator, compare approximate values of old and new forms. If secure, test a new factor-form curve with unequal roots and a non-unit coefficient. The check samples only a small part of the quadratic-function sub-topic.

## Optional swaps · worked results

| Day | Swap 1 | Swap 2 |
|---|---|---|
| 1 | `sqrt(25)=5`; `sqrt(7)` surd and `2<sqrt(7)<3` | `sqrt(49)=7`; `sqrt(15)` surd and `3<sqrt(15)<4` |
| 2 | `sqrt(108)=sqrt(36×3)=6sqrt(3)`; square gives 108 | `sqrt(200)=sqrt(100×2)=10sqrt(2)`; square gives 200 |
| 3 | `sqrt(18)=3sqrt(2)`, `sqrt(72)=6sqrt(2)`; difference `3sqrt(2)` cm | `sqrt(27)=3sqrt(3)`, `sqrt(75)=5sqrt(3)`; difference `2sqrt(3)` cm |
| 4 | `sqrt(20)+sqrt(45)=2sqrt(5)+3sqrt(5)=5sqrt(5)` cm | `sqrt(28)+sqrt(63)=2sqrt(7)+3sqrt(7)=5sqrt(7)` units |
| 5 | `sqrt(75)-sqrt(27)=5sqrt(3)-3sqrt(3)=2sqrt(3)` cm, positive | `sqrt(125)-sqrt(20)=5sqrt(5)-2sqrt(5)=3sqrt(5)` cm |
| 6 | `sqrt(8)sqrt(18)=sqrt(144)=12` cm² | `(3sqrt(2))(2sqrt(10))=6sqrt(20)=12sqrt(5)` |
| 7 | `5/sqrt(3)=5sqrt(3)/3` | `6/sqrt(7)=6sqrt(7)/7`, both ≈2.2678 |
| 8 | `(sqrt(50)+sqrt(8))/sqrt(2)=(5sqrt(2)+2sqrt(2))/sqrt(2)=7` | `(sqrt(75)-sqrt(12))/sqrt(3)=(5sqrt(3)-2sqrt(3))/sqrt(3)=3` |
| 9 | `(−1,2)`, `x=−1`, `(0,3)`, no real zeros since y≥2 | `(1,3)`, `x=1`, `(0,5)`, no real zeros since y≥3; x=0 and x=2 both yield y=5 |
| 10 | Zeros 2,6; axis x=4; vertex `(4,−4)`; y-intercept `(0,12)`, opens up | Zeros −2,4; axis x=1; vertex `(1,9)`; y-intercept `(0,8)`, opens down; at x=1 y=9>0 |

**Scope:** these key values support feedback on initial surd skills and first quadratic features. They do not cover quadratic formula, completing the square, discriminant, full curve sketching or modelling. School assessment design remains with the school.