Advice on ${4-{x-2\over x+2}}$

Question

On the weekend I help out maths at a club for students of 15-16 years of age. I did a survey on a particular algebra question. I asked all of them roughly 200 students(not in a sitting but spread out the entire days) to simplify $\color{red}{4-{{{{x-2\over x+2}}}}}$ into a single fraction. I collected all the answers and counted them.

I found out that 70% of the students got it wrong!

They gave $\color{blue}{3x+6\over x+2}$ instead of $\color{green}{3x+10\over x+2}$

Where do you think they went wrong? What advice of teaching techniques do you advice tutors to do, to help teach them better, so in the future they won't get it wrong again in these kind of basic algebraic fractions questions?

Answer 1

1. Tell them to decompose the computation

$$4-\frac{x-2}{x+2}=\frac{4(x+2)-(x-2)}{x+2}$$ instead of rushing to

$$\frac{4x+8-x-2}{x+2}.$$

2. Make them check with numerical instances.

With $x=2$, $4$ vs. $\dfrac{12}4$.

Answer 2

Possibly done mistake like: $${4-{x-2\over x+2}} = \frac{4x+8-x-2}{x+2} = \frac{3x+6}{x+2}$$

Instead of $${4-{x-2\over x+2}} = \frac{4x+8-x+2}{x+2} = \frac{3x+10}{x+2}$$

Answer 3

Distributing the negative sign to the numerator before doing addition would help

$$-\frac{x-2}{x+2}=\frac{2-x}{x+2}$$