Wir üben die Falltechnik auf der Weichbodenmatte.

Because the main noun in the compound is Technik, and Technik is feminine in German: die Technik.

In German compound nouns, the last part determines the gender. So:

• der Fall = fall
• die Technik = technique
• die Falltechnik = falling technique / breakfall technique

Here it is also the direct object of üben, but feminine singular looks the same in nominative and accusative:

• nominative: die Falltechnik
• accusative: die Falltechnik

So the form stays die.

Üben means to practice.

It normally takes a direct object in the accusative, just like to practice something in English:

• Wir üben die Falltechnik. = We practice the falling technique.
• Ich übe Deutsch. = I practice German.
• Sie übt den Handstand. = She practices the handstand.

So in this sentence, die Falltechnik is what is being practiced.

Falltechnik is a compound noun:

• Fall = fall
• Technik = technique

In sports, martial arts, gymnastics, or physical education, Falltechnik usually means a technique for falling safely or a breakfall technique.

A learner might wonder whether Fall means grammatical case here, because Fall can also mean that in grammar. But in this sentence, because of the sports context and the mat, it clearly means falling.

Because auf is a two-way preposition. It can take:

• dative for location: where something happens
• accusative for direction/movement toward a destination

Here, the sentence describes where the practice takes place: on the soft mat. That is location, so German uses the dative:

• auf der Weichbodenmatte = on the soft mat

Compare:

• Wir üben auf der Weichbodenmatte. = We practice on the mat.
• Wir gehen auf die Weichbodenmatte. = We go onto the mat.

So der shows location, not movement onto it.

Der is not only masculine nominative. It can also be the dative singular form for feminine nouns.

The noun here is die Matte (feminine), so:

• nominative: die Weichbodenmatte
• accusative: die Weichbodenmatte
• dative: der Weichbodenmatte

Because auf expresses location here, it requires the dative, which is why you get der Weichbodenmatte.

It is a large soft gym mat, often used in schools, gymnastics, martial arts, or movement training.

The word is another compound:

• weich = soft
• Boden = floor / ground
• Matte = mat

So Weichbodenmatte is basically a soft floor mat or thick landing mat.

German often creates long compound nouns like this instead of using separate words.

Yes, it is very normal.

German main-clause word order usually puts the conjugated verb in second position:

• Wir = subject
• üben = verb
• die Falltechnik = direct object
• auf der Weichbodenmatte = prepositional phrase of place

So the sentence structure is:

Subject + verb + object + location

That is a very natural order in German.

Yes, that is possible, but it sounds more marked.

Putting auf der Weichbodenmatte earlier gives a little more attention to the location. The more neutral version is:

• Wir üben die Falltechnik auf der Weichbodenmatte.

The alternative is not wrong, but it may sound like you are emphasizing where you practice it.

Because weich is not being used as a separate adjective in the sentence. It is part of a compound noun.

Compare:

• die weiche Matte = the soft mat
  Here weiche is an adjective and gets an ending.

• die Weichbodenmatte = the soft-floor mat
  Here Weichbodenmatte is one single noun, so weich does not take an adjective ending.

This is very common in German compounds.

In this sentence, wir simply means we.

German wir is the normal first-person plural pronoun. It does not have the extra meanings that English you can sometimes have. It refers to the speaker plus at least one other person.

So Wir üben ... means that the speaker is included in the group doing the practice.

Yes. Then you would use the plural die Falltechniken:

• Wir üben die Falltechniken auf der Weichbodenmatte.

In the plural:

• singular: die Falltechnik
• plural: die Falltechniken

The article die stays the same, but the noun changes to -en in the plural.