How can interactions with A.I. systems be designed? This paper explores the design space for A.I. interaction to develop tools for designers to think about tangible and physical A.I. interactions. Our proposed framework consists of two dimensions: initiative (human, mixed, or machine) and materiality (physical, combined, or digital form). A particularly interesting area of interactions we identify is the quadrant of physical, machine-initiated interactions. With our framework, we examine calculator interactions and attempt to expand these to the tangible, mixed-initiative space. We illustrate each area in our proposed framework with one representative example of a calculator -- a common and well-known example of a computing device. We discuss existing examples of calculators and speculative future interactions with mixed-initiative and physical calculator systems. We reflect on the implications of our framework for the larger task of designing human-A.I. collaborative systems. Designers can also apply this framework as a guideline for analogous solutions to problems in the same domain.