7.1 Identify two mathematical errors in the learner's reasoning and correct each one. 1.7.2 Write corrective feedback suitable for a Grade 6 learner. Your feedback must include one concrete representation or teaching move, such as a table, graph, number line, or balance approach.

Here are the solutions for questions 1.7.1 and 1.7.2:

1.7.1 Identify two mathematical errors in the learner's reasoning and correct each one.

• Error 1: Misunderstanding of Proportionality

  • Learner's reasoning: "Because this is a straight-line graph, it is proportional."
  • Correction: A relationship is proportional only if its graph is a straight line that passes through the origin $(0,0)$. The cost function $C(n) = 6n + 30$ is a straight line, but it does not pass through the origin (it passes through $(0,30)$). Therefore, it is a linear relationship, but not a proportional one.

• Error 2: Arithmetic Error in Solving the Equation

  • Learner's reasoning: The learner correctly reached $4n = 80$ but then incorrectly concluded $n=5$.
  • Correction: To solve $4n = 80$, you must divide both sides by 4. $$4n = 80$$ $$\frac{4n}{4} = \frac{80}{4}$$ $$n = \boxed{20}$$ The learner incorrectly divided 20 by 4, or made an arithmetic error in dividing 80 by 4.

1.7.2 Write corrective feedback suitable for a Grade 6 learner. Your feedback must include one concrete representation or teaching move, such as a table, graph, number line, or balance approach.

"Great job setting up the equation to find the number of packs! Let's look at two things.

First, about straight lines and proportionality: A straight line is only proportional if it starts at zero. Think about the cost function $C(n) = 6n + 30$. If you print 0 packs, the cost is R30 (the setup fee). Since the cost isn't R0 when you have 0 packs, it's a straight line, but not a proportional one.

Second, when you had $4n = 80$, you need to find what one 'n' is. Imagine you have a balance scale:

$$\text{Left side: } 4 \times n \quad \text{Right side: } 80$$

To find out what one 'n' is, you need to divide both sides by 4 to keep the scale balanced:

$$\frac{4n}{4} = \frac{80}{4}$$

So, $n$ should be $\boxed{20}$, not 5. This means the club needs to sell 20 packs to make R50 profit."