ASRS 2048404

During an instructional flight, after performing a touch and go, climbing in the pattern, we turned left crosswind. After completing the turn, the student and I visually identified traffic ahead and to our right, but very close. Due to the closeness of the two, I directed the student to turn downwind which kept lateral separation from the other aircraft, but changed the sequence of our arrival in the pattern. I contacted the tower, and they told the other (outside) aircraft to perform a 360. After listening to the tower recording, and viewing the ADS-B data, I'm aware of several factors and mistakes that led to the loss of separation. My focus was on evaluating my student's performance. While I was observing traffic visually and using an ADS-B connected iPad, I lost sight and awareness of the aircraft in the pattern ahead of us. Being seated on the right side of the aircraft, and making a left climbing turn, it was not possible for me to see the other aircraft until we completed the crosswind turn. I am now aware that the other aircraft extended upwind, possibly out of confusion based on what I heard in replaying the audio. Had ATC instructed them to extend upwind, I would have been alerted to the need for us to delay our turn. I incorrectly assumed that they were flying a normal pattern, and we did not need to delay our turn to crosswind. I visually checked to our left before making the turn, but the other aircraft was farther ahead, and not visible to me. Once we turned crosswind, and I identified the other aircraft, I made the decision to turn inside of them. At the moment, it seemed that trying to maneuver behind them would have been a more difficult maneuver. Ideally, we should not have turned crosswind, but given that we did, I can also see that returning to upwind, or just making a turn to the right would have kept us in the proper sequence. I'm still concerned that making a right turn might have brought us closer to the other aircraft which would have created a greater hazard.