Distributes Over…

The “distributive property” of multiplication and division.  That’s not the proper full name, but it’s what many people say… so, when do you distribute, and when don’t you?  That is the question.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

$\dfrac{(4)(3)(2k+6)}{6}$

$=\dfrac{(12)(8k+24)}{6}$

$=\dfrac{(12)(8k+24)}{(2)(3)}$

$=\dfrac{(6)(4k+12)}{3}$

$=(2)(4k+4)$

$=8k+4$

.

..

Hint 1: The correct answer to the original problem is: $4k+12$

Hint 2: The complete description of the distributive property of multiplication is: “the distributive property of multiplication over…”?

Hint 3: Several mistakes have been made in the work above – don’t just seek the correct answer – look for mistakes between every pair of lines.