Flight Instructor with student reported a NMAC during pattern entry at a non-towered airport. The other aircraft took evasive action to avoid a collision.

Narrative

While conducting a flight lesson my student departed ZZZ on Rwy XXL after completing a few touch and go landings and flew straight out. The student leveled out at 2200 feet to set up for a left downwind entry to [the] RWY at ZZZ1. There was an aircraft climbing out next to us on our right at the same altitude. As we leveled off we kept them in sight to ensure it was clear to pass underneath them. The tower also confirmed that our right turn was approved and cleared us to change frequencies. The student began the turn to the right to fly towards the the left downwind for ZZZ1. The student was not sure of the frequency for ZZZ1 so I asked them how they could find it. This was my second flight with the student who asked me to help them prepare for a checkride; so I wanted to see how the student was going to handle this situation. Instead of turning downwind he stopped the turn and started heading northwest towards the approach end. He asked me for the frequency and I asked them to use one of their resources to find the frequency. The student got frustrated that I would not give them the answer and proceeded to say that they couldn't look up the frequency because they are flying the plane. After it was clear that they were not going to look it up using the EFB or the GPS; I began to show them how to find it using the Airports page on the GPS. When I looked up and to the right; I saw an airplane in a right bank maneuvering around us about 500 away at the same altitude. It was clear that other airplane took evasive action to avoid us. We were still on the ZZZ radio frequency so I do not know if they were trying to make radio contact with us. My assumption is the other plane departed ZZZ1 and I did not see them as we flew towards the northwest. My eyes were inside the cockpit after being caught up in the situation and lost situational awareness at a crucial time.