I had also learnt my first two groups of broken plurals at around the same time I posted the above comments on sound plurals.
So, qalam (pen) becomes aqlaam (pens); bayt (house) becomes buyoot (houses).
Two things I found particularly interesting:
1. Arabic sometimes makes loan words fit into one of the broken plural patterns. So film becomes aflaam (films); bank becomes bunook (banks). English similarly adapts its loan words - we tend to say 'referendums' and 'stadiums' in everyday speech (not referenda, stadia). We definitely say 'dachshunds' and not Dachshunde.
2. The concept of broken plurals is strange to me, an English speaker, but it kinda introduced me to Arabic tri-literal roots. The three root letters of, say, qalam is q,l,m and this is kept in the plural. The loan word film is treated as if it has three root letters (f,l,m) which are kept in the plural, aflaam, and in the same order.
And now for something very logical yet alien to me at the same time: the root letters have a basic meaning and new words can be made from them by changing vowels inbetween and sometimes adding prefixes and suffixes but the root letters must remain in the same order.
The root letters d,r,s have the meaning of studying. This is shown in the simple past tense: darasa (he studied), darastu (I studied), darasta/i (you studied) ... in the present tense: adrus (I study), tadrus (you study), etc. ... and in the noun dars (lesson). The prefix ma means 'place' so madrasa means school, i.e place of study. The prefix mu means the one doing the verb so mudarris means teacher, i.e. one who causes others to study. Pretty neat, huh?
Just have to give one more example - I'm geeking out here! XD
Root letters k,t,b (writing) gives kataba (he wrote), katabtu (I wrote), kitaab (book), maktab (office/desk) and maktaba (library). I don't know the word for 'librarian' but I expect it to be something like mukaatib.
I can remember only a few more off the top of my head - e.g. khubz (bread) and khubbaaz (baker); muhaasib (accountant) and hisaab (bank account) - but I know matbakh means kitchen. It must mean place where one cooks so t,b,kh must mean cooking. Gotta be, hasn't it?