Research studies have found that mathematics teachers face challenges when designing technology-integrated lessons to support mathematical inquiry with authentic problems. This suggests that teachers need to develop technological pedagogical knowledge (TPACK) or their professional knowledge for technology integration. Drawing upon the concept of TPACK, this study defines teachers' professional knowledge for mathematics technology integration as technological pedagogical mathematical knowledge (TPMK) and examines the different kinds of TPMK that are needed to support the design of technology-integrated lessons for mathematical inquiry with authentic problems. Using content analysis, the design talk of three mathematics teachers designing a series of technology-integrated lessons to teach the concept of average through the inquiry of authentic problems was examined to articulate their associated TPMK. Thematic analysis was then used to derive five pathways used by the teachers to create the TPMK for mathematical inquiry with authentic problems throughout the design meetings that the teachers held across 2 months. These pathways articulate the process of teachers' pedagogical decision-making as they create TPMK. The implications of these findings for future development of teacher knowledge for technology-supported mathematics inquiry are discussed.