Flight Instructor reported an NMAC event during landing pattern training. Flight Instructor took control of aircraft and executed an evasive maneuver and avoided a collision.

Narrative

I was conducting training at ZZZ; The lesson we were conducting involved performing traffic pattern operations; On our third lap in the pattern after completing a 2nd touch and go Tower advised another aircraft to take off (Aircraft Y); we reached 800 AGL and began turning our crosswind. While on upwind I was trying to find the departing aircraft behind but couldn't get visual contact with them out of the rear window. We reached traffic pattern altitude of 1100 ft. on the crosswind and began turning crosswind to downwind. In the turn I looked down to check my IPAD to see if I could gain some situational awareness to where the traffic on upwind was; to my surprise as well as towers the traffic began an early turn to crosswind and was only -200 ft. below and less than a 1/4 mile on a collision course with. Seeing this I took flight controls from my student added power and pitched up to gain as much altitude as possible. Tower noticed the unexpected turn and asked the other airplane if it had us in sight because they had turned into our path in the pattern. The other aircraft continued to climb and advised looking for us as they passed underneath our aircraft by maybe 100 ft.