This article describes some of the essential mathematics that underpins the use of algorithms through a series of learning pathways. To begin, a graphic depicting the mathematical ideas and concepts that underpin the learning of algorithms for multiplication and division is provided. The understanding and use of algorithms is informed by two important ideas--grid multiplication and extended multiplication facts. The graphic combines a number of learning pathways that lead to those two ideas. The article goes on to demonstrate how each element builds part of the underpinning structure needed to understand algorithms and to use them efficiently.