According to the Common Core State Standards Initiative, arithmetic at grade 5 focuses on several goals, the first of which is the continued practice of the addition and subtraction of fractions. Multiplication and division of fractions are also introduced at this time, though the fraction units -- numerators and denominators -- at this level are all whole numbers. Division in grade 5 is extended to two-digit divisors, and operations involving decimals are solved to hundredths. Grade 5 is also when understanding of volume is developed, along with algebraic thinking and basic graphing.
Expressions and equations are introduced in grade 6, as are rates and ratios, all of which involve multiplication and division of whole numbers. The number system is extended to now include negative numbers. Multiplication and division of fractions is completed, and the concepts of functions, geometry, statistics and probability are introduced.
In grade 7, the skills introduced in grade 6 are developed further and more specifically. Expressions and linear equations are used to solve both real-life and mathematical problems. Geometry is applied to scale drawings and three-dimensional constructions, and is used to solve problems of volume, area and surface area. The understanding of proportional relationships is another goal of grade 7 math, as is the use of statistics to draw inferences regarding population.
Grade 8 marks the time when irrational numbers are first acknowledged. Expressions and equations now include radicals and integer exponents. Students are guided through the relationships between linear equations, lines and proportions. The Pythagorean Theorem is applied to geometry, and problems involving the volume of cylinders, spheres, and cones is introduced. Bivariate data, or data for two variables, is also covered, and students will learn how to explore patterns of association.