Flight Instructor reported a NMAC while in the traffic pattern at a congested airport.

Narrative

Delayed start for first familiarization flight for private pilot student. Schedule for XA:15 depart out of class C for basic training. By the time we got to the runup area it was obvious that accomplishing those goals during remaining daylight would not be possible. Asked the tower if they could accommodate remaining in pattern of the shorter runway. The pattern was quite full; but the tower tried to accommodate. Cleared for take off to follow 172 ahead. Took off with 172 in sight. Without positively verifying 172 had turned downwind; turned crosswind. Tower advised of close proximity of 172 which passed under us at estimated 300.' Tower vectored us back into downwind to follow aircraft and cleared for the option. Overshot base to final of smaller parallel runway. Tower correctly directed us to land or depart area. Landed. Briefed student on lessons learned - relearned. They are well understood by this instructor. The best solution was to have cancelled flight:Contributing Factors: Slow is fast; that is take whatever time to do things right or fast will be problematic. There was insufficient time to reasonably accomplish this first flight. Daylight flight became compressed based on the late start. At this point especially with new student; flight should have been cancelled in favor of ground review. Flight instructor should set the example; in this case I did not. The runway was into the setting sun. This was another flag as both upwind and downwind visibility was compromised by setting sun low angle. This seems to have amplified the haze issue. Changing plan on short notice; away from what was briefed from departing class C to remaining in the pattern. Although patterns have been briefed and flown in ATD; this was not the appropriate time to go straight into pattern work. Additionally there were probably 5 aircraft working the pattern at this high volume training facility. It was not a constructive learning environment for this student. Performance of RV in terms of speed and climb rate significantly exceed that of a 172 and should have lead proactively to power/airspeed/climb reduction to match the prevailing characteristics of the aircraft in the pattern. HF (Human Factors) Decisions: Compressed schedule warranted flight cancellation. Changing flight plan on short notice was inappropriate. Given student's experience; remaining in the pattern in class C space was an error. Simply too high workload to devote proper attention to student. Aircraft is equipped with the latest GARMIN technology including traffic displays. Should have set the 750 to traffic display with short range. This is a key factor.