teaching this critical area is the first content standard. All students must be taught to "understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they relate to one another; [and] compute fluently and make reasonable estimates." The ability to perform mental calculations and to calculate answers on paper is "essential." Algebra: The PSSM names four skills related to algebra that should be taught to all students: to "understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; [and] analyze change in various contexts." Very simple algebra skills are often taught to young children. For example, a student might convert an addition equation such as 19 + 15 = ? into a simpler equation, 20 + 14 = ? for easy calculation. Formally, this is described in algebraic notation like this: (19 + 1) + (15 − 1) = x, but even a young student might use this technique without calling it algebra. The PSSM recommends that all students complete pre-algebra coursework by the end of eighth grade and take an algebra class during high school. Geometry: The overall goals for learning geometry are to "analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; specify locations and describe spatial relationships using coordinate geometry and other representational systems; apply transformations and use symmetry to analyze mathematical situations; [and] use visualization, spatial reasoning, and geometric modeling to solve problems." Some geometry skills are used in many everyday tasks, such as reading a map, describing the shape of an object, arranging furniture so that it fits in a room, or determining the amount of fabric or construction materials needed for a project. Teaching should be appropriate to students' developmental level: Young students should be able to explain the difference between a rectangle and a square, while older students should be able to express more complex reasoning, including simple mathematical proofs. (See van Hiele model.) The PSSM promotes the appropriate use of physical objects, drawings, and computer software for teaching geometry. Measurement: Measurement skills have many practical applications, as well as providing opportunities for advancing mathematical understand and for practicing other mathematical skills, especially number operations (e.g., addition or subtraction) and geometry. Students should "understand measurable attributes of objects and the units, systems, and