The purpose of this article was to describe the task design and implementation of cultural artefacts in a mathematics lesson based on the integration of modelling and conjecturing perspectives. The conceived process of integrating a soccer ball into mathematics lessons via modelling- and conjecturing-based instruction was first detailed. Next, the paper analysed six students' participation behaviours as they created mathematical problems, definitions, terms, representations and arguments during modelling and conjecturing activities. Findings suggested students effectively engaged in the search for soccer ball models and solutions to posed questions, especially the reason manufacturers prefer soccer balls constructed from regular pentagons and regular hexagons to other types of regular polygons.