One of the things I, and probably all math teachers, struggle with is teaching students when they can and cannot distribute. For example, students learn very early on in their mathematical education that:

One reason we know this is valid is because we can calculate the answer using the rules of BEDMAS and obtain the same final answer:

This pattern extends to variables as well. Often it takes students a few tries, but eventually they become proficient in questions like:

However, the problem begins in grade 10. Students begin to encounter expressions like . They believe that the same pattern should hold. And so, they dutifully write:

Next, they apply this pattern of distributing incorrectly to square roots:

Where clearly the answer should be:

And once grade 11 and 12 hit, the number of bad distributing examples proliferates!

Ultimately, I think we as math teachers are to blame for the problem. Since distributing is so natural for us, we gloss over how unique and remarkable it actually is. As such, we give students a cavalier attitude towards distributing and they happily apply it in every context they encounter. As the above list suggests, distributing is very rare and only applies in a vary narrow set of situations. When teaching the distributive property, we should do a better job demonstrating just how unusual this property is.