折れる and 折る: a snapped branch and folded paper
Distinguish a long object snapping from folding paper, and compare the results expressed by 折った and 折れた.
When a branch snaps
雪の重みで枝が折れました。 says that a branch snapped under the weight of snow. おれる describes the change with the branch as its subject.
枝を折らないでください。 asks someone not to break a branch. 折る, read おる, is a godan verb and becomes 折らない in the negative.
Folding is not necessarily damage
紙を半分に折ってください。 asks someone to fold paper in half. 折る also describes making a fold, so it does not always mean snapping something.
折った紙を封筒に入れました。 means that someone put the folded paper in an envelope. What 折った describes depends on whether its noun is, for example, a branch or paper.
Distinguish 折った from 折れた
鉛筆の芯が折れたので、削りました。 says that the pencil lead broke, so the speaker sharpened the pencil. 折れた is the past form of 折れる and does not imply deliberate action.
折る has the past form 折った, while 折れる has 折れた. Learn 紙を折った and 芯が折れた with their particles and objects, rather than relying only on their similar sounds.
Which requests folding paper in half: 紙を半分に折ってください or 紙が半分に折れてください?
Show the answer and why
紙を半分に折ってください is correct. The person addressed acts on the paper, so use 紙を and 折って, the te-form of 折る.