Sipho expands 3(x + 4) as 3x + 4. Which response gives the best diagnosis and feedback?

Question 3:
D. The learner multiplied $x$ by 3 but not 4; use equal groups or an area model to obtain $3x + 12$.

Handwritten evidence:
Let $x=2$.
Sipho's expression: $3x+4 = 3(2)+4 = 6+4 = 10$.
Correct expansion: $3(x+4) = 3(2+4) = 3(6) = 18$.
Since $10 \neq 18$, Sipho's expansion is incorrect.
The misconception is that Sipho only applied the multiplication to the first term inside the parenthesis ($x$) and not to the second term (4). The distributive property requires multiplying the outside term by every term inside the parenthesis.

Question 4:
C. Kabelo's method preserves equality and gives $x=6$; Lerato failed to multiply 4 by 3 when removing the brackets.

Handwritten evidence:
Original equation: $3(x+4) = 30$

Check Lerato's value $x = \frac{26}{3}$:

$$3\left(\frac{26}{3}+4\right) = 3\left(\frac{26}{3}+\frac{12}{3}\right) = 3\left(\frac{38}{3}\right) = 38$$

Since $38 \neq 30$, Lerato's value is incorrect.

Check Kabelo's value $x = 6$:

$$3(6+4) = 3(10) = 30$$

Since $30 = 30$, Kabelo's value is correct.

Kabelo's method is sound because it correctly applies inverse operations to both sides of the equation, maintaining equality. Lerato's method is flawed because she incorrectly applied the distributive property, failing to multiply 4 by 3.

Question 5:
B. Pages copied are the input; total cost is the output because cost depends on the number of pages.

Handwritten evidence:
The two quantities are:

1. Number of pages copied ($p$)
2. Total cost ($C$)

The direction of dependence is that the total cost ($C$) depends on the number of pages copied ($p$). The more pages copied, the higher the total cost. Therefore, the number of pages copied is the independent variable (input), and the total cost is the dependent variable (output).