Mathematical models and expressions traditionally evolved as symbolic representations, with cognitively arbitrary rules of symbol manipulation. The embodied mathematics philosophy posits that abstract math concepts are layers of metaphors grounded in our intuitive arithmetic capabilities, such as categorizing objects and part-whole analysis. We introduce a design framework that facilitates the construction and exploration of embodied representations for algebraic expressions, using interactions inspired by innate arithmetic capabilities. We instantiated our design in a sketch interface that enables construction of visually interpretable compositions that are directly mappable to algebraic expressions and explorable through a ladder of abstraction. The emphasis is on bottom-up construction, with the user sketching pictures while the system generates corresponding algebra. We present diverse examples created by our prototype system. A coverage of the US Common Core curriculum and playtesting studies with children point to the future direction and potential for a sketch-based design paradigm for mathematics.