I leveled up my Badger and MG18A by doing the dailies, took me a couple of weeks or so just playing on one character. Remember to up the difficulty before you start, you get double materials.
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Does setting heroic difficulty for any pre-HoW strike automatically provide materials?
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Not sure about the strikes but on the daily heroic stories, the hardest difficulty (usually 30 I think?) gives two ascendant materials and the difficulty below gives you one. I need an odd number so the last one was a run-through on 28 or 28 to get one ascendant shard - job done :D