Using the examples that you have given, this is my opinion of how the sentences should read, (in British English).
"On the last Friday of every month, his Maths teacher, Mr Randal, gives a test to his students".
"Last Friday ... " refers to only ONE Friday! It means the Friday of last week; the Friday of the week which has recently passed; the Friday which has just gone. "Last Friday ... " cannot refer to several Fridays!
The words, "Mr. Randal", are in apposition to "his Maths teacher", so should be set within commas, to act as �brackets �.
The Teacher "gives a test", (or "sets a test"). The Student "takes a test", (or "sits a test").
" ... students with bad marks in the test ... "
The most usual preposition is "in".
The other Teachers are correct in what they write.
Two men were travelling together through a wood when suddenly they met a bear on their way. One of them, who happened to be in front, climbed quickly up a tree and hid himself in the branches. The other, seeing no help and scared that he must be attacked, fell flat on the ground, with his face in the dust.
"Two men were travelling together through a wood, when suddently, they met a bear".
"On their way" is superfluous --- How else, "suddenly", could they "meet a bear"? if they were NOT "on their way? "On their way" should be omitted!
"One of them, who was in front, climbed quickly up a tree and hid in the branches", or more probably:"The one who was in front climbed quickly up a tree and hid in the branches".
I have no problem with the phrase "in front", because it means "in a position just ahead of or further forward than someone or something else". (Oxford Dictionary of English)
What relevance are the words: " ... who happened to be in front ..."? What is so special about being "in front"? What befell him, because he was "in front"? Nothing!. So these words, " ... happened to be ..." can be omitted. The man simply was "in front".
" ... hid himself in the branches." When we omit the word "himself" does it alter the sense of the sentence and make us think that he may have hidden his friend in the branches? No! So, we can omit "himself" without altering the sense of the text.
"The other, seeing no help coming, and scared that he might be attacked, fell flat to the ground, with his face in the dust".
"... seeing no help coming ..." The man was unable to see any assistance, actually arriving, which would help him.
" ... and scared that he might be attacked". He was afraid of the possibility of attack, ("might"), not the necessity of being attacked, which is what "must"implies.
" ... fell flat to the ground ...". This means that he deliberately threw himself to the ground, to avoid detection. The phrase " ... fell flat on the ground ...", means that he accidently fell to the ground.
"The dog was eating the last of the meat".
This means that several portions of meat had been eaten previously, and that this was the VERY LAST portion. After the dog ate the meat, there was none left. "The last of" The only part of something that remains: They drank the last of the wine. (Oxford Dictionary of English).
I hope that I have been able to help you.
Les