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The participles are verbal adjectives. Being adjectives, they have gender, number, and case; and hke other adjectives they agree in gender, number, and case with the nouns that they modify. On the other hand, since they partake of the nature of verbs, (a) they have tense and voice, (b) they receive, like other parts of a verb, adverbial modifiers, and (c) if they be participles of a transitive verb they can take a direct object.
Examples:
(1) ὁ ἀπόστολος λέγων ταῦτα ἐν τῷ ἱερῷ βλέπει τὸν κύριον, the apostle, saying these things in the temple, sees the Lord
Here the participle λέγων, which means saying, agrees with ἀπόστολος, which is in the nominative case and singular number and is a masculine noun. The participle, therefore, must be nominative singular masculine. On the other hand, the participle is enough of a verb to have tense and voice. It is in the present tense because the action which it denotes is represented as going on at the same time as the action of the leading verb βλέπει; it is in the active voice because it represents the apostle as doing something, not as having something done to him. And it has the adverbial modifier ἐν τῷ ἱερῷ and the direct object ταῦτα. On the other hand, it has no subject, as a finite verb (e.g. an indicative) would have; for the noun ἀπόστολος, which denotes the person represented as performing the action denoted by the participle, is not the subject of the participle, but the noun with which the participle, like any other adjective, agrees.
(2) βλέπομεν τὸν ἀπόστολον λέγοντα ταῦτα ἐν τῷ ἱερῷ, we see the apostle saying these things in the temple
Here the noun with which the participle agrees is accusative singular masculine. Therefore the participle must also be accusative singular masculine. But its direct object and its adverbial modifier are the same as in (1).
(3) προσερχόμεθα τῷ ἀποστόλῳ λέγοντι ταῦτα ἐν τῷ ἱερῷ, we come to the apostle while he is saying these things in the temple
Here the participle λέγοντι agrees with a masculine noun in the dative singular and must therefore itself be dative singular masculine. But in this example it is quite impossible to translate the participle literally. The translation, we come to the apostle saying these things in the temple, would not do at all, for in that English sentence the participle saying would be understood as agreeing not with the apostle but with the subject of the sentence, we. It is necessary, therefore, to give up all attempts at translating the participle "literally". Instead, we must express the idea which is expressed by the Greek participle in an entirely different way - by the use of a temporal clause. When such temporal clauses are used to translate a Greek present participle they are usually introduced by while. Such a free translation would have been better than the literal translation even in Example (1), although there the literal translation was not absolutely impossible. It would have been rather better to translate ὁ ἀπόστολος λέγων ταῦτα ἐν τῷ ἱερῷ βλέπει τὸν κύριον by while the apostle is saying these things in the temple, he sees the Lord.
(4) διδασκομένῳ ὑπὸ τοῦ ἀποστόλου προσέρχονται αὐτῷ οἱ δοῦλοι, while he is being taught by the apostle, the servants are coming to him
Here διδασκομένῳ agrees with αὐτῷ, which, like τῷ ἀποστόλῳ in the preceding example, is dative with the verb προσέρχονται. διδασκομένῳ is the present passive participle of διδάσκω.