These are fresh fictional cases relative to cards and optional swaps. The teacher-targeted worked key is publicly accessible. Collect an initial attempt before showing it; the tasks are not secure exams or QCAA assessment instruments. Every route below requires the same exact values, reasons and checks.
Check A · Day 5 fresh surd transfer
An invented display has two straight paper strips, a longer one of length sqrt(243) cm and a shorter one of length sqrt(48) cm. These are constructed lengths for mathematics practice, not measured objects.
- Simplify both lengths using square factors and state the exact difference, longer minus shorter.
- A draft says the difference is
sqrt(195)cm because243-48=195. Explain the invalid rule with an exact comparison. - Estimate your exact difference to one decimal place. State which part of your response is exact and which is approximate.
Routes: write the factorisation and explanation; give a spoken derivation while someone records exact notation and your reason; use square-factor tiles to arrange both lengths and dictate the exact difference and error diagnosis.
Check B · Day 10 fresh graph and surd transfer
The abstract rule y=-(x-2)(x-8) is invented for this check. Separately, a pure numerical ratio is 5/sqrt(11); it has no units and is not part of the graph model.
- Find both x-intercepts, the axis of symmetry, turning point, y-intercept and opening direction of the graph. Show at least one substitution or sign check.
- State where the graph is above the x-axis and justify using the factor signs or your sketch.
- Rationalise
5/sqrt(11)exactly. Explain why the denominator becomes rational and how you know the value has not changed.
Routes: write steps and a labelled sketch; explain aloud with coordinates, interval and radical equality captured by a scribe; use labelled point/axis and fraction tiles then dictate the same mathematical reasoning.