This article presents an examination of the language demands of cognitively demanding tasks and proposes an initial framework for the language demands of higher-order mathematics thinking practices. We articulate four categories for this framework: "language of generalisation," "language of comparison," "language of proportional reasoning," and "language of analysing impact." These categories were developed out of our collaborative work to design and implement higher-order thinking tasks with a group of Grade 9 (14- and 15-year-olds) teachers teaching in a linguistically diverse setting; analyses of student work samples on these tasks; and our knowledge of the literature. We describe each type of language demand and then analyse student work in each category to reveal linguistic challenges facing students as they engage these mathematical tasks. Implications for teaching and professional development are discussed.