CFI INSTRUCTIONAL IN A PIPER AZTEC IS CLRED TO OTHER LGB RWY AND COMES INTO CONFLICT WITH A C150 ON APCH TO SAME RWY.

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FLT TRAINING WAS BEING CONDUCTED IN A PIPER AZTEC AT LONG BEACH ARPT. THE LCL CTLR (FREQ OF 119.4) WAS HANDLING THE L PATTERN FOR RWY 25L AND RWY 30 ARRS. THIS WAS THE CTLR THAT WAS PROVIDING SVC TO US AS WE WERE DOING 'STOP AND GO' CLOSE PATTERN OPS FOR RWY 25L. DURING THIS TIME; OPS WERE BEING CONDUCTED FOR RWY 25R; AND WAS BEING HANDLED BY A SECOND LCL CTLR ON FREQ OF 120.5. THE PATTERN FOR RWY 25R IS R-HAND TFC. THE LCL CTLR (119.4) ASKED US TO FOLLOW A FASTER AIRPLANE ENTERING THE PATTERN FOR RWY 25L. AS WE TURNED FROM XWIND TO DOWNWIND; WE STARTED FOLLOWING THIS AIRPLANE. THE PRECEDING AIRPLANE WAS THEN ASKED TO EXTEND HIS DOWNWIND FOR SEQUENCING. THIS MADE US EXTEND ON THE DOWNWIND AS WELL. THE LCL CTLR (119.4) NEVER GAVE US FURTHER INSTRUCTIONS AT THIS POINT. SO; I INSTRUCTED MY STUDENT TO TELL HIM THAT WE WERE DOWNWIND WITH THE PRECEDING AIRPLANE STILL IN SIGHT. THE NEXT SET OF INSTRUCTIONS GIVEN TO US WERE TO EXTEND DOWNWIND AND CROSS THE 605 FREEWAY (APPROX 5 MI E OF THE ARPT) FOR BASE TO RWY 25R. HAVING BEEN SIDESTEPPED BEFORE UNDER SIMILAR CONDITIONS AT LGB ARPT; I DID NOT THINK ANYTHING DIFFERENT ABOUT THIS INSTRUCTION. WE CONTINUED AS INSTRUCTED; HOWEVER; THE LCL CTLR DID NOT INSTRUCT US TO SWITCH TO THE OTHER LCL CTLR OF 120.5. I MADE 2 ATTEMPTS TO VERIFY RWY ASSIGNMENT ON THE NOW CONGESTED FREQ 119.4. MY CALLS WERE TO VERIFY AND TO GET A FREQ CHANGE AS NEEDED. WE THEN STARTED OUR BASE TURN ONTO FINAL FOR RWY 25R. 2 EVENTS HAPPENED AT THIS SAME TIME: 1) THE LCL CTLR ASKED US TO 'MAKE A R TURN OVER THE 605.' 2) MY STUDENT SAW A SINGLE ENG CESSNA AT ABOUT 200 FT OFF OUR R WING TURNING ONTO FINAL FROM THE R PATTERN. WE HAD TO DEVIATE FROM ATC INSTRUCTIONS BY MAKING A STEEP L TURN TO AVOID TFC. CONTRIBUTING FACTORS INCLUDE VARIED MIX OF ACFT IN THE PATTERN; FREQ CONGESTION; AND TOO MANY ACFT BEING HANDLED AT ONE TIME. FACTORS AFFECTING HUMAN PERFORMANCE WERE TRYING TO UNDERSTAND VAGUE INSTRUCTIONS FROM ATC; RADIO CONGESTION; LONG DOWNWIND IN HAZY VISIBILITY. CLASS C AIRSPACE MAY REDUCE THE LCL CTLR'S WORKLOAD BY NOT HAVING TO SEQUENCE ARRIVING VFR ACFT INTO THE PATTERN.