Which order includes cockroaches and praying mantises?

The order that includes both cockroaches and praying mantises is Blattodea. This is because Blattodea is specifically designated for cockroaches, and it also encompasses the termite suborder, which reflects the close evolutionary relationship between these insects. The distinction of Blattodea is crucial in understanding the classification, as it highlights the unique characteristics and biological attributes that these insects share.

On the other hand, the other options represent different insect orders. Orthoptera includes grasshoppers and crickets, which are not related to cockroaches or mantises. Hemiptera consists of true bugs, such as aphids and cicadas, which have completely different lifecycles and characteristics. Coleoptera, known as the order of beetles, encompasses a large variety of beetle species that are distinct in morphology and behavior from those found in Blattodea. Thus, recognizing the correct classification of cockroaches and praying mantises under Blattodea is vital for understanding entomology and insect taxonomy.