### 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.

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.

### NAEP Mathematics Objectives – Mathematical Content Areas

#### Number Properties and Operations

Number sense is a major expectation of the 2007 NAEP. At fourth grade, students are expected to have a solid grasp of whole numbers, as represented by the decimal system, and to have the beginnings of understanding fractions. By eighth grade, they should be comfortable with rational numbers, represented either as decimal fractions (including percents) or as common fractions. They should be able to use them to solve problems involving proportionality and rates. Also in middle school, number should begin to coalesce with geometry via the idea of the number line. This should be connected with ideas of approximation and the use of scientific notation. Eighth graders should also have some acquaintance with naturally occurring irrational numbers, such as square roots and pi. By 12th grade, students should be comfortable dealing with all types of real numbers.

 GRADE 12 1) Number sense a) Write, rename, represent, or compare real numbers (e.g.,ï  pi , numerical relationships using number lines, models, or diagrams. b) Represent very large or very small numbers using scientific notation in meaningful contexts. c) Find or model absolute value or apply to problem situations. d) Interpret calculator or computer displays of numbers given in scientific notation. e) Order or compare real numbers, including very large or small real numbers. 2) Estimation a) Establish or apply benchmarks for real numbers in contexts. b) Make estimates of very large or very small numbers appropriate to a given situation by: • identifying when estimation is appropriate or not, • determining the level of accuracy needed, • selecting the appropriate method of estimation, or • analyzing the effect of an estimation method on the accuracy of results. c) Verify solutions or determine the reasonableness of results in a variety of situations including scientific notation, calculator, and computer results. d) Estimate square or cube roots of numbers less than 1,000 between two whole numbers. 3) Number operations a) Perform computations with real numbers including common irrational numbers or the absolute value of numbers. b) Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a real number by: • zero, or • a number less than zero, or • a number between zero and one, or • one, or • a number greater than one. c) Solve application problems involving numbers, including rational and common irrationals, using exact answers or estimates as appropriate. 4) Ratios and proportional reasoning a) Use proportions to model problems. b) Use proportional reasoning to solve problems (including rates). c) Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships). 5) Properties of number and operations a) Solve problems involving factors, multiples, or prime factorization. b) Use prime or composite numbers to solve problems. c) Use divisibility or remainders in problem settings. d) Apply basic properties of operations. e) Provide a mathematical argument about a numerical property or relationship.

#### Measurement

In this NAEP Mathematics Framework, attributes such as capacity, weight/mass, time, and temperature are included, as well as the geometric attributes of length, area, and volume. Although many of these attributes are included in the grade 4 framework, the emphasis is on length, including perimeter, distance, and height. More emphasis is placed on area and angle in grade 8. By grade 12, volumes and rates constructed from other attributes, such as speed, are emphasized.

Units involved in items on the NAEP assessment include non-standard, customary, and metric units. At grade 4, common customary units such as inch, quart, pound, and hour and the common metric units such as centimeter, liter, and gram are emphasized. Grades 8 and 12 include the use of both square and cubic units for measuring area, surface area, and volume; degrees for measuring angles; and constructed units such as miles per hour. Converting from one unit in a system to another (such as from minutes to hours) is an important aspect of measurement included in problem situations. Understanding and using the many conversions available is an important skill. There are a limited number of common, everyday equivalencies that students are expected to know.

Items classified in this content area depend on some knowledge of measurement. For example, an item that asks the difference between a 3-inch and a 1¾-inch line segment is a number item, while an item comparing a 2-foot segment with an 8-inch line segment is a measurement item. In many secondary schools, measurement becomes an integral part of geometry; this is reflected in the proportion of items recommended for these two areas.

 GRADE 12 1) Measuring physical attributes a) Compare objects with respect to length, area, volume, angle measurement, weight, or mass. b) Estimate the size of an object with respect to a given measurement attribute (e.g., area). c) Select or use appropriate measurement instrument to determine or create a given length, area, volume, angle, weight, or mass. d) Solve mathematical or real-world problems involving perimeter or area of plane figures such as or composite figures. e) Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes. f) Solve problems involving indirect measurement such as finding the height of a building by comparing its shadow with the height and shadow of a known object. g) Solve problems involving rates such as speed or population density. 2) System of measurement a) Select or use appropriate type of unit for the attribute being measured such as length, area, angle, time, or volume. b) Solve problems involving conversions within the same measurement system such as conversions involving square inches and square feet. c) Estimate the measure of an object in one system given the measure of that object in another system and the approximate conversion factor. For example: • Distance conversion: 1 kilometer is approximately e of a mile. • Money conversion: U.S. dollar is approximately 1.5 Canadian dollars. • Temperature conversion: Fahrenheit to Celsius d) Determine appropriate size of unit of measurement in problem situation involving such attributes as length, area, or volume. e) Determine appropriate accuracy of measurement in problem sit­uations (e.g., the accuracy of each of several lengths needed to obtain a specified accuracy of a total length) and find the measure to that degree of accuracy. f) Construct or solve problems (e.g., floor area of a room) involving scale drawings.

#### Geometry

By grade 4, students are expected to be familiar with a library of simple figures and their attributes, both in the plane (lines, circles, triangles, rectangles, and squares) and in space (cubes, spheres, and cylinders). In middle school, understanding of these shapes deepens, with the study of cross-sections of solids and the beginnings of an analytical understanding of properties of plane figures, especially parallelism, perpendicularity, and angle relations in polygons. Right angles and the Pythagorean theorem are introduced, and geometry becomes more and more mixed with measurement. The basis for analytic geometry is laid by study of the number line. In high school, attention is given to Euclid’s legacy and the power of rigorous thinking. Students are expected to make, test, and validate conjectures. Via analytic geometry, the key areas of geometry and algebra are merged into a powerful tool that provides a basis for calculus and the applications of mathematics that helped create the modern technological world in which we live.

Symmetry is an increasingly important component of geometry. Elementary students are expected to be familiar with the basic types of symmetry transformations of plane figures, including flips (reflection across lines), turns (rotations around points), and slides (translations). In middle school, this knowledge becomes more systematic and analytical, with each type of transformation being distinguished from other types by their qualitative effects. For example, a rigid motion of the plane that leaves at least two points fixed (but not all points) must be a reflection in a line. In high school, students are expected to be able to represent transformations algebraically. Some may also gain insight into systematic structure, such as the classification of rigid motions of the plane as reflections, rotations, translations, or glide reflections, and what happens when two or more isometries are performed in succession (composition).

 GRADE 12 1) Dimension and shape a) Use two-dimensional representations of three-dimensional objects to visualize and solve problems involving surface area and volume. b) Give precise mathematical descriptions or definitions of geometric shapes in the plane and in three-dimensional space. c) Draw or sketch from a written description plane figures (e.g., isosceles triangles, regular polygons, curved figures) and planar images of three-dimensional figures (e.g., polyhedra, spheres, and hemispheres). d) Describe or analyze properties of spheres and hemispheres. 2) Transformation of shapes and preservation of properties a) Recognize or identify types of symmetries (e.g., point, line, rotational, self-congruences) of two- and three-dimensional figures. b) Give or recognize the precise mathematical relationship (e.g., congruence, similarity, orientation) between a figure and its image under a transformation. c) Perform or describe the effect of a single transformation on two- and three-dimensional geometric shapes (reflections across lines of symmetry, rotations, translations, and dilations). d) Describe the final outcome of successive transformations. e) Justify relationships of congruence and similarity, and apply these relationships using scaling and proportional reasoning. 3) Relationships between geometric figures a) Apply geometric properties and relationships in solving multi-step problems in two and three dimensions (including rigid and non-rigid figures). b) Represent problem situations with geometric models to solve mathematical or real-world problems. d) Use the Pythagorean theorem to solve problems in two- or three-dimensional situations. c) Describe and analyze properties of circles (e.g., perpendicularity of tangent and radius, angle inscribed in a semicircle). d) Analyze properties or relationships of triangles, quadrilaterals, and other polygonal plane figures. e) Describe or analyze properties and relationships of parallel, perpendicular, or intersecting lines, including the angle relationships that arise in these cases. 4) Position and direction a) Describe the intersection of two or more geometric figures in the plane (e.g., intersection of a circle and a line). b) Visualize or describe the cross section of a solid. c) Represent geometric figures using rectangular coordinates on a plane. d) Use vectors to represent velocity and direction. 5) Mathematical reasoning a) Make, test, and validate geometric conjectures using a variety of methods including deductive reasoning and counterexamples.

#### Data Analysis and Probability

Students in grade 12 will be expected to use a wide variety of statistical techniques for all phases of the data analysis process, including a more formal understanding of statistical inference (but still with simulation as the main inferential analysis tool). In addition to comparing univariate data sets, students at this level should be able to recognize and describe possible associations between two variables by looking at two-way tables for categorical variables or scatter plots for measurement variables. Association between variables is related to the concepts of independence and dependence, and an understanding of these ideas requires knowledge of conditional probability. These students should be able to use statistical models (linear and non-linear equations) to describe possible associations between measurement variables and should be familiar with techniques for fitting models to data.

 GRADE 12 1) Data representation Histograms, line graphs, scatter plots, box plots, circle graphs, stem and leaf plots, frequency distributions, and tables. a) Read or interpret data, including interpolating or extrapolating from data. b) For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, scatter plots, line graphs). c) Solve problems by estimating and computing with univariate or bivariate data (including scatter plots and two-way tables). d) Given a graph or a set of data, determine whether information is represented effectively and appropriately (bar graphs, box plots, histograms, scatter plots, line graphs). e) Compare and contrast the effectiveness of different representations of the same data. 2) Characteristics of data sets a) Calculate, interpret, or use mean, median, mode, range, interquartile range, or standard deviation. b) Recognize how linear transfor­mations of one-variable data affect mean, median, mode, and range (e.g., effect on the mean by adding a constant to each data point). c) Determine the effect of outliers on mean, median, mode, range, interquartile range, or standard deviation. d) Compare two or more data sets using mean, median, mode, range, interquartile range, or standard deviation describing the same characteristic for two different populations or subsets of the same population. e) Given a set of data or a scatter plot, visually choose the line of best fit and explain the meaning of the line. Use the line to make predictions. f) Use or interpret a normal distri­bution as a mathematical model appropriate for summarizing certain sets of data. g) Given a scatter plot, make decisions or predictions involving a line or curve of best fit. h) Given a scatter plot, estimate the correlation coefficient (e.g., Given a scatter plot, is the cor­relation closer to 0, .5, or 1.0? Is it a positive or negative correlation?). 3) Experiments and samples a) Identify possible sources of bias in data collection methods and describe how such bias can be controlled and reduced. b) Recognize and describe a method to select a simple random sample. c) Make inferences from sample results. d) Identify or evaluate the charac­teristics of a good survey or of a well-designed experiment. 4) Probability a) Analyze a situation that involves probability of independent or dependent events. b) Determine the theoretical probability of simple and compound events in familiar or unfamiliar contexts. c) Given the results of an experiment or simulation, estimate the probability of simple or compound events in familiar or unfamiliar contexts. d) Use theoretical probability to evaluate or predict experimental outcomes. e) Determine the number of ways an event can occur using tree diagrams, formulas for combinations and permutations, or other counting techniques. f) Determine the probability of the possible outcomes of an event. g) Determine the probability of independent and dependent events. h) Determine conditional probabil­ity using two-way tables. i) Interpret probabilities within a given context.

#### Algebra

In high school, students should become comfortable in manipulating and interpreting more complex expressions. The rules of algebra should come to be appreciated as a basis for reasoning. Non-linear functions, especially quadratic functions, and also power and exponential functions, are introduced to solve real-world problems. Students should become accomplished at translating verbal descriptions of problem situations into symbolic form. Expressions involving several variables, systems of linear equations, and the solutions to inequalities are encountered by grade 12.

 GRADE 12 1) Patterns, relations, and functions a) Recognize, describe, or extend arithmetic, geometric progressions, or patterns using words or symbols. b) Express the function in general terms (either recursively or explicitly), given a table, verbal description, or some terms of a sequence. c) Identify or analyze distinguishing properties of linear, quadratic, inverse (y = k/x) or exponential functions from tables, graphs, or equations." d) Determine the domain and range of functions given various contexts. e) Recognize and analyze the general forms of linear, quadratic, inverse, or exponential functions (e.g., in y = ax + b, recognize the roles of a and b). f) Express linear and exponential functions in recursive and ex­plicit form given a table or verbal description. 2) Algebraic representations a) Translate between different rep­resentations of algebraic expressions using symbols, graphs, tables, diagrams, or written ­descriptions. b) Analyze or interpret relationships expressed in symbols, graphs, tables, diagrams, or written descriptions. c) Graph or interpret points that are represented by one or more ordered pairs of numbers on a rectangular coordinate system. d) Perform or interpret transformations on the graphs of linear and quadratic functions. e) Use algebraic properties to develop a valid mathematical argument. f) Use an algebraic model of a situation to make inferences or predictions. g) Given a real-world situation, determine if a linear, quadratic, inverse, or exponential function fits the situation (e.g., half-life bacterial growth). h) Solve problems involving exponential growth and decay. 3) Variables, expressions, and operations a) Write algebraic expressions, equations, or inequalities to represent a situation. b) Perform basic operations, using appropriate tools, on algebraic expressions (including grouping and order of multiple operations involving basic operations, ex­ponents, roots, simplifying, and ­expanding). c) Write equivalent forms of algebraic expressions, equations, or inequalities to represent and explain mathematical relationships. 4) Equations and inequalities a) Solve linear, rational, or quad­ratic equations or inequalities. b) Analyze situations or solve problems using linear or quadratic equations or inequalities sym­bolically or graphically. c) Recognize the relationship between the solution of a system of linear equations and its graph. d) Solve problems involving more advanced formulas [e.g., the volumes and surface areas of three dimensional solids; or such formulas as: A = P(1 + r)t, A = Pert]." e) Given a familiar formula, solve for one of the variables. f) Solve or interpret systems of equations or inequalities.