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# National Standards for Grade 12 - Mathematics (page 3)

— National Assessment Governing Board
Updated on Mar 14, 2011

#### 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.