Why use “ing” in word playing?

Question

Why is -ing used in this sentence in the word Playing and participating?. The tense in this sentence is not present continuous,so why use -ing?

Our youngest kids needed athletic shoes for playing outside and for participating in gym class at school.

Answer 1

The -ing forms in this sentence are not used as components of tensed verbs in the continuous construction but as nouns, objects of the preposition for.

Our kids needed athletic shoes.
What did they need them for?
They needed them for playing outside and for participating in gym class.

When an -ing form is used this way, as a noun, it is called a gerund. Gerunds act like both nouns and verbs. As verbs they take subjects and objects and are modified by adverbials; but as nouns they (or the clauses which they head) may serve as subjects or objects in a clause or as objects of prepositions.

Swimming is fun. .. In this sentence the gerund swimming is the subject.

Playing football is fun. ... In this sentence, the gerund playing takes the object football, but the gerund phrase playing football is the subject of the sentence. (In traditional grammar, the gerund playing would be regarded as the subject, but that is a minor detail).

We need a ball for playing football. .. In this sentence the gerund phrase playing football (or the gerund playing) is the object of the preposition for.

Answer 2

The words "playing" and "participating" in your sentence are actually gerunds. The gerund is a verb form that acts as a noun. In fact, gerunds can even serve as the subject of a sentence:

Playing outside is really fun. [gerund]

The gerund and present participle both have the same form, but they serve different purposes. As you imply in your question, the present participle can be used as part of the present continuous tense:

Our youngest kids need athletic shoes because they are playing outside and are participating in gym class at school. [present continuous]

Note that the present continuous is formed using "[conjugated form of 'to be'] + [present participle]", which distinguishes it from the gerund form.

Answer 3

To add some more introductory information on the -ing forms. Here is an excerpt from Practical English Usage by Michael Swan,

293 -ing forms (1): introduction
1 'participles' and 'gerunds'

We can use -ing forms (e.g. smoking, walking) not only as verbs, but also like adjectives or nouns.
Compare:
You're smoking too much these days. (verb: part of present progressive)
There was a smoking cigarette end in the ashtray. (adjective describing cigarette end)
Smoking is bad for you. (noun: subject of sentence)

When -ing forms are used as verbs or adjectives, they are often called 'present participles'. (This is not a very suitable name, because these forms can refer to the past, present or future.) When they are used more like nouns, they are often called 'gerunds'. In Practical English Usage the expression '-ing form' is used except when there is a good reason to use one of the other terms.