Zanele uses 𝑓(𝑥) = 𝑥2 −6𝑥 +11 to connect a rule, a table and a graph. Which description is consistent for 𝑥 = 2, 3 and 4? A. The outputs are 3, 2, 3; the graph has a maximum (3, 2) because every squared expression opens downwards. B. The outputs are 3, 2, 3; the graph has a minimum (3, 2) and a symmetry line 𝑥 = 3. C. The outputs are 4, 9, 16; the graph is 𝑦 = 𝑥2 because the linear and constant terms may be ignored. D. The outputs are 3, 2, 1; the first difference stays at −1, so the graph continues as a straight line. Handwritten evidence: Evaluate the function at 𝑥 = 2, 3 and 4, then use the results to justify the vertex and line of symmetry. Question 18 Sipho plans a Grade 6 lesson on a growing tile pattern. Which sequence most strongly develops generalisation across representations? A. Learners copy a diagram, a table and a graph, but the teacher treats each representation as a separate exercise with no connecting questions. B. The teacher selects an attractive real-world story, supplies every count and rule, and asks learners to decorate the final diagram before discussing the mathematics. C. The teacher states the rule first, demonstrates substitution, and asks learners to calculate twenty values without returning to the pattern. D. Learners build early terms, record counts in a table, compare what changes and what stays fixed, propose a rule, and explain how the table and diagram agree. Handwritten evidence: Apply two explicit criteria, identify one we

We substitute each value of $x$ into the function $f(x) = x^2 - 6x + 11$.
For $x = 2$:
$f(2) = (2)^2 - 6(2) + 11 = 4 - 12 + 11 = 3$
For $x = 3$:
$f(3) = (3)^2 - 6(3) + 11 = 9 - 18 + 11 = 2$
For $x = 4$:
$f(4) = (4)^2 - 6(4) + 11 = 16 - 24 + 11 = 3$
The outputs for $x = 2, 3, 4$ are $3, 2, 3$.

Based on the outputs $3, 2, 3$:

• Option A states outputs are 3, 2, 3, but incorrectly claims a maximum and that the graph opens downwards. A quadratic with a positive $x^2$ coefficient (like $x^2 - 6x + 11$) opens upwards, having a minimum.
• Option B states outputs are 3, 2, 3, and correctly identifies a minimum (3, 2) and a symmetry line $x = 3$. This is consistent with the calculated outputs.
• Option C states incorrect outputs (4, 9, 16) and an incorrect simplification of the function.
• Option D states incorrect outputs (3, 2, 1) and an incorrect conclusion about the first difference and graph type.

Therefore, option B is the consistent description.

The outputs are $f(2)=3$, $f(3)=2$, and $f(4)=3$.
The lowest output value is 2, which occurs when $x=3$. For a parabola that opens upwards (which $f(x) = x^2 - 6x + 11$ does, as the coefficient of $x^2$ is positive), this lowest point is the vertex.
Thus, the vertex of the parabola is $(3, 2)$.
The line of symmetry of a parabola passes through its vertex. Since the vertex is at $x=3$, the line of symmetry is $x=3$. This is further supported by the symmetric outputs: $f(2)=3$ and $f(4)=3$, which are equidistant from $x=3$.

The final answer is $\boxed{\text{B}}$

Question 2: Sipho plans a Grade 6 lesson on a growing tile pattern. Which sequence most strongly develops generalisation across representations? A. Learners copy a diagram, a table and a graph, but the teacher treats each representation as a separate exercise with no connecting questions. B. The teacher selects an attractive real-world story, supplies every count and rule, and asks learners to decorate the final diagram before discussing the mathematics. C. The teacher states the rule first, demonstrates substitution, and asks learners to calculate twenty values without returning to the pattern. D. Learners build early terms, record counts in a table, compare what changes and what stays fixed, propose a rule, and explain how the table and diagram agree. Handwritten evidence: Apply two explicit criteria, identify one we

Option D most strongly develops generalization across representations.

Two explicit criteria for strong generalization across representations:

1. Active Learner Construction and Discovery: Learners are actively involved in building the pattern (diagrammatic representation), recording data (tabular representation), analyzing changes, and formulating a rule (algebraic representation). This hands-on approach fosters deeper understanding and ownership of the generalization process.
2. Explicit Connection and Integration of Multiple Representations: The activity requires learners to move between different representations (diagram, table, rule) and, crucially, to "explain how the table and diagram agree." This explicit linking helps learners see the underlying mathematical structure across various forms, which is key to generalization.

One weak criterion (identified from other options):

1. Treating Representations in Isolation: As seen in Option A, if different representations (diagram, table, graph) are presented as separate exercises without explicit connecting questions, learners will struggle to see the relationships between them, hindering their ability to generalize across these representations.