### Mathematics achievement levels at grade 12

The following mathematics achievement levels describe what 12th-graders should know and be able to do in mathematics at each level.

#### Basic:

Twelfth-grade students performing at the Basic level should be able to solve mathematical problems that require the direct application of concepts and procedures in familiar situations. For example, they should be able to perform computations with real numbers and estimate the results of numerical calculations. These students should also be able to estimate, calculate, and compare measures and identify and compare properties of two- and three-dimensional figures, and solve simple problems using two-dimensional coordinate geometry. At this level, students should be able to identify the source of bias in a sample and make inferences from sample results, calculate, interpret, and use measures of central tendency and compute simple probabilities. They should understand the use of variables, expressions, and equations to represent unknown quantities and relationships among unknown quantities. They should be able to solve problems involving linear relations using tables, graphs, or symbols; and solve linear equations involving one variable.

#### Proficient:

Students in the twelfth grade performing at the Proficient level should be able to select strategies to solve problems and integrate concepts and procedures. These students should be able to interpret an argument, justify a mathematical process, and make comparisons dealing with a wide variety of mathematical tasks. They should also be able to perform calculations involving similar figures including right triangle trigonometry. They should understand and apply properties of geometric figures and relationships between figures in two and three dimensions. Students at this level should select and use appropriate units of measure as they apply formulas to solve problems. Students performing at this level should be able to use measures of central tendency and variability of distributions to make decisions and predictions; calculate combinations and permutations to solve problems, and understand the use of the normal distribution to describe real-world situations. Students performing at the Proficient level should be able to identify, manipulate, graph, and apply linear, quadratic, exponential, and inverse proportionality (y = k/x) functions; solve routine and non-routine problems involving functions expressed in algebraic, verbal, tabular, and graphical forms; and solve quadratic and rational equations in one variable and solve systems of linear equations.

#### Advanced:

Twelfth-grade students performing at the Advanced level should demonstrate in-depth knowledge of the mathematical concepts and procedures represented in the framework. They can integrate knowledge to solve complex problems and justify and explain their thinking. These students should be able to analyze, make and justify mathematical arguments, and communicate their ideas clearly. Advanced level students should be able to describe the intersections of geometric figures in two and three dimensions, and use vectors to represent velocity and direction. They should also be able to describe the impact of linear transformations and outliers on measures of central tendency and variability; analyze predictions based on multiple data sets; and apply probability and statistical reasoning in more complex problems. Students performing at the Advanced level should be able to solve or interpret systems of inequalities; and formulate a model for a complex situation (e.g., exponential growth and decay) and make inferences or predictions using the mathematical model.

Reprinted with the permission of the National Assessment Governing Board.

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