Proofs are attempts to conclusively demonstrate the validity of the claim for all cases indicated within its domain, which implies that proving should involve thoughtful consideration of the domain. This study analyzed the enactment of three general claim tasks, or tasks where the domain of the claim referred to an infinite number of cases, that were used during an introduction-to-proof teaching experiment with 10 ninth grade students. We analyzed the tasks in terms of the opportunities students experienced to engage in reasoning-and-proving and attend to the domain of the claims. The use of general claim tasks provided students with opportunities to engage in varied reasoning-and-proving activities, including forms not typically found in textbooks. Students' attention to the domain of the claims increased over the course of study as a result of the teacher-researcher's continued focus on this aspect of the tasks, although their attention did not always encompass all cases within the domain. By making the domain of mathematical claims a central focus, we emphasize its important role in the reasoning-and-proving opportunities afforded to students and contribute to an understanding of students' early interpretations of this aspect of proof tasks.