I explore students' discourses in small groups working on mathematical problems using GeoGebra, focusing on the Cartesian connection between algebra and geometry. Specifically, the interest lies in what is internally persuasive for students in upper-secondary school (11th grade) with histories of low attainment. Three problem-solving episodes are presented in detail to illuminate different intertwining and interacting discourses that students produce. In all cases, students engaged in visual explorations and expressed joy in the work. Students used discourses of visual appearance and technical symbolism to talk about screen objects and problem solutions. Two cases illustrate how a student makes a conjecture internally before verifying it for themselves and then convincing their peers of the validity of their solution. In a contrasting case, students used visual trial and error and asked an authority to confirm their solution. Discourses concerning the institutional demands of school interfered with students' mathematical dialogue.