Mathematics questions at grades 4 and 8 are based on the five content areas that are listed below. At grade 12, because of changes made to the framework in 2005, the content areas of measurement and geometry were combined (see "Mathematics Framework Changes" for further information).
Number Properties and Operations
This content area focuses on students' abilities to represent numbers, order numbers, compute with numbers, make estimates appropriate to given situations, use ratios and proportional reasoning, and apply number properties and operations to solve real-world and mathematical problems. This content area also addresses number sense—comfort in dealing with numbers—and addresses students' understanding of what numbers tell us, equivalent ways to represent numbers, and the use of numbers to represent attributes of real-world objects and quantities. At grade 4, the focus is on whole numbers and fractions; at grade 8, the focus extends to include rational numbers; at grade 12, the focus extends to include real numbers.
This content area focuses on students' understanding of measurement attributes such as capacity, weight/mass, time, and temperature as well as the geometric attributes of length, area, and volume. Students may be asked to select appropriate units and tools for measuring, to measure length with a ruler at all three grades, to measure angles with a protractor at grades 8 and 12, and to solve application problems related to units of measurement. At grade 4, the focus is on length, including perimeter, distance, and height. At grades 8 and 12, students are also expected to understand and demonstrate knowledge of volume and surface area. Knowledge of both customary and metric units is expected. Students may be asked to solve problems that require conversions between (with conversion factors given) or within systems of measurement.
This content area focuses on identification of geometric shapes and transformations and combinations of those shapes. By grade 4, students are expected to be familiar with simple plane figures such as lines, circles, triangles, and rectangles as well as solid figures such as cubes, spheres, and cylinders. They are also expected to be able to recognize examples of parallel and perpendicular lines. As students move to middle school and beyond, increased understanding should deepen of two- and three-dimensional figures, especially parallelism, perpendicularity, angle relations in polygons, congruence, similarity, and the Pythagorean theorem. Students at all grades are expected to show knowledge of symmetry and transformations of shapes and to identify images resulting from flips, rotations, or turns. Justifications and reasoning in both formal and informal settings are expected at grades 8 and 12.
Data Analysis and Probability
This content area focuses on students’ skills in four areas: data representation, characteristics of data sets, experiments and samples, and probability. At grade 4, students are expected to use standard statistical measures such as the median, range, or mode, and to compare sets of related data; at grades 8 and 12, they are also expected to show understanding of other statistical concepts such as the impact of outliers and the line of best fit in a scatterplot. By grade 8, students are expected to have some knowledge of experiments and samples, such as being able to recognize possible sources of bias in sampling and identify random versus nonrandom sampling; by grade 12, students are also expected to make inferences from sample results. Students at all grades are expected to use statistics and statistical concepts to analyze and communicate interpretations of data. Students may be asked to solve problems that address appropriate methods of gathering data, the visual exploration of data, ways to represent data, or the development and evaluation of arguments based on the analysis of data. Probability is assessed informally at grade 4 and more formally at grades 8 and 12.
This content area focuses on students’ understanding of patterns, relations, and functions; algebraic representation; variables, expressions, and operations; and equations and inequalities. At grade 4, students are expected to show knowledge of simple patterns and expressions; at grade 8, this knowledge extends to include linear equations; and at grade 12, it extends further to include quadratic and exponential equations and functions. Representational skills, such as students’ ability to translate between different forms of representation (e.g., from a written description to an equation), the ability to graph and interpret points located on a coordinate system, and the ability to use algebraic properties to draw a conclusion, are assessed in this area. Students may be asked to express relationships algebraically as number sentences, equations, or inequalities; manipulate algebraic expressions; or solve and interpret algebraic equations and inequalities that are grade-level appropriate.
Learn more about the mathematics assessment.