In a biased weak (a,b) polyform achievement game, the maker and the breaker alternately mark a,b previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a,b) game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all (a,b) pairs for polyiamonds and polyominoes up to size four.