To address the upcoming reform curriculum initiative following competency-based learning in Vietnam, this study examines how variations of a contextualized task allow students to experience the mathematical modelling process. We explore how Grade 10 students respond to tasks with different levels of authenticity when they first attempt such tasks. Adapting the Theory of Authentic Task Situations, we designed variations of the Goat Cage task and implemented them with six small groups of students. Results showed that the modelling processes that the students progressed through differed in mathematical models used and real-world considerations, with diverse approaches when solving the more-authentic variation. Not all phases of the modelling process were observed explicitly when they engaged in the tasks. Students showed a positive disposition towards tasks with more authentic contextual information. The study suggests ways to adapt contextualized tasks to make them more authentic, which allows students to experience a holistic modelling process.