Ramadan: When praying in Jama'ah and you only reach the last raka'ah...?

For example, last night, my sister was praying Isha and I decided to pray with her. But by the time I joined her she was in the last raka'ah. Thus I had to make up 3 raka'ahs. So in the raka'ah I prayed with her - I read al-fatiha only. Then she did the tahiyat and her salams. Then I got up and prayed... show more For example, last night, my sister was praying Isha and I decided to pray with her. But by the time I joined her she was in the last raka'ah. Thus I had to make up 3 raka'ahs. So in the raka'ah I prayed with her - I read al-fatiha only. Then she did the tahiyat and her salams. Then I got up and prayed another raka'ah and only read fatiha again. Then I did the tahiyat. Then I got up again and read fatiha + another surah, and again i did another raka'ah with fatiha and another surah. then my tahiyat, then my salams. To summarize it, after my sister did her salams, I prayed:


Rakah 1 = fatiha
TAHIYAT
Raka'ah 2 = fatiha + another surah
Raka'ah 3 = fatiha + another surah
TAHIYAT

I basically prayed the salah backwards. Is this correct? If not, how are you supposed to do it?
JazakAllahu Khairan
Update: Oh dear.. I think some of you have misunderstood. By "Jama'ah" I meant.. ONLY my sister and I were praying together. At first she was praying alone, then I decided to join her instead of praying on my own so that I could get more good deeds. You get it? And according to my knowledge, that's... show more Oh dear.. I think some of you have misunderstood. By "Jama'ah" I meant.. ONLY my sister and I were praying together. At first she was praying alone, then I decided to join her instead of praying on my own so that I could get more good deeds. You get it? And according to my knowledge, that's allowed. It wasn't as though she was leading men. Sorry if I phrased it wrong before.
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