Present Participle

See general explanation here

In the picture:
διαλεγόμεναι ἀλλήλαις ὑφαίνουσιν.
The main action is ὑφαίνουσιν: they are weaving
the participating action is διαλεγόμεναι αλλήλαις while talking to each other.
We could replace the present participle construction by a different pattern altogether:
      ἐν ᾦ δὲ ὑφαίνουσιν, διαλέγονται ἀλλήλαις. (while they weave, they chat to one another)
This second construction is what we tend to use, the Greeks and the Romans preferred the former. So, if we want to learn Greek properly, we've got to get used to participle constructions as well.

Like the Greeks we say:
I stop working:  παύω ἐργαζόμενος 
But we can also use the infinitive (the to do form) in English:
   I see him come
   I hear her leave
In Greek we can't, we've got to say:
   ὁρῶ αὐτὸν ἐρχόμενον 
   ἀκούω αὐτῆς ἀπερχομένης (Remember that ἀκούω is followed by the genitive case form)

Exercises

For reference, see the full declension table in your book, pages 84/85 (116 second edition)
Then go over exercise 8α page 85 (8β 2nd edition), trying to visualize the situations.
When you're ready for it, try 8β (8γ 2nd edition). But again, visualize the action first.

Then change things around a bit:
Instead of one man, imagine several, or women.

Instead of boys fighting see girls going for one another, or athletes, dogs, oxen. Just translating the sentences and applying the rules is not good enough. Because it is not applied to real life and experience, and therefore dead theory only. And if you think that you get good at a language through theory and grammar exercises, there's a nasty surprise in store for you: You won't! Witness the millions of children and grown-ups who've learnt their theory and word-lists diligently and when given the opportunity to use the language (or even only read it) are hopelessly lost. They've learnt to swim on dry land, learnt to ski on the living-room carpet.
 So when it says (exercise 8b, sentence 6)
εἰς τὴν νῆσον πορευόμενοι οἱ ἄνθρωποι πολλὰ καὶ δεινὰ πάσχουσιν. 
you must see them sailing those seven seas suffering all kinds of hardships: storms, sickness, lack of food, foul drinking water, and the sea still stretching as far as the eye can see, endless. Where is that blessed island?