Avia per scalas lente ambulat, et puer prope gradus eam exspectat.

Because avia is the subject of ambulat. In Latin, the subject is usually in the nominative case, and avia is nominative singular.

If you had aviam, that would be accusative singular, which is usually used for a direct object, not the subject.

So:

• avia ambulat = grandmother walks
• puer aviam exspectat = the boy waits for grandmother

Per is a preposition that takes the accusative case, so scalas is accusative plural.

The phrase per scalas means something like along/on/by way of the stairs. In natural English, that is often expressed simply as up the stairs, down the stairs, or on the stairs, depending on the context.

So the important grammar point is:

• per
  • accusative

Latin often uses the plural scalae/scalas for a flight of stairs, much like English usually says stairs rather than a stair.

So per scalas is very natural Latin for movement involving a staircase.

Yes, they overlap, but there is a small difference in feel:

• scalae / scalas often refers to the staircase or stairs as a whole
• gradus often refers more to the individual steps

So in this sentence, per scalas suggests movement on the staircase, while prope gradus suggests being near the steps.

Latin often has pairs of words like this that are close in meaning but not identical.

Because lente is an adverb, meaning slowly, and it modifies the verb ambulat.

• lente ambulat = she walks slowly

By contrast, lenta is an adjective meaning slow, feminine singular, and it would describe a noun:

• avia lenta = a slow grandmother

So:

• adjective = describes a noun
• adverb = describes a verb

Yes. Ambulat is present tense, third person singular, from ambulare.

It can mean either:

• she walks
• she is walking

Latin does not usually make the same sharp distinction that English does between simple present and present continuous.

Because puer is in the nominative singular, which is the normal case for the subject, and exspectat is third person singular, so they match.

Also, eam is accusative, so that shows it is the object, not the subject.

So the structure is:

• puer = subject
• eam = object
• exspectat = verb

Here gradus is accusative plural, because it follows prope, and prope as a preposition takes the accusative.

That can be confusing because gradus is a fourth-declension noun, and its accusative plural looks the same as its nominative singular:

• nominative singular: gradus
• accusative plural: gradus

So in this sentence, the meaning and the preposition prope tell you that it must be accusative plural.

Eam means her. It is the accusative singular feminine form of the pronoun is, ea, id.

It refers back to avia.

So:

• avia = grandmother
• eam = her = grandmother

Latin often uses a pronoun like this when English would also say her.

Because the Latin verb exspectat takes a direct object in the accusative.

So Latin says:

• eam exspectat = literally he awaits her

In natural English, we usually say he waits for her, but Latin does not need a separate word for for here.

This is a very common thing to notice when moving from English to Latin: a verb may use a direct object in Latin even if English uses a preposition.

Yes. Latin word order is much more flexible than English word order because the case endings show what each word is doing.

For example, these would still be understandable:

• Puer eam prope gradus exspectat.
• Prope gradus puer eam exspectat.

The meaning stays basically the same because:

• puer is nominative
• eam is accusative

Still, Latin word order is not random. Authors move words around for emphasis, rhythm, or style. In this sentence, the order is natural and clear.

Because Classical Latin does not have articles like English the and a/an.

So avia can mean:

• grandmother
• a grandmother
• the grandmother

Which one is best depends on context. The same is true for puer.

Yes, those spellings are closely related, and you may see both.

• exspectat is a standard classical spelling
• expectat is also very common

Most beginner textbooks prefer exspectat, but if you later see expectat, it is not a different verb in meaning.