# Pennsylvania State Standards for Mathematics: Grade 12

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PA.2.1.11. Numbers, Number Systems and Number Relationships: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.1.11.A. Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots, finding logarithms).

PA.2.2.11. Computation and Estimation: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.2.11.A. Develop and use computation concepts, operations and procedures with real numbers in problem-solving situations.

2.2.11.B. Use estimation to solve problems for which an exact answer is not needed.

2.2.11.C. Construct and apply mathematical models, including lines and curves of best fit, to estimate values of related quantities.

2.2.11.D. Describe and explain the amount of error that may exist in a computation using estimates.

2.2.11.E. Recognize that the degree of precision needed in calculating a number depends on how the results will be used and the instruments used to generate the measure.

2.2.11.F. Demonstrate skills for using computer spreadsheets and scientific and graphing calculators.

PA.2.3.11. Measurement and Estimation: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.3.11.A. Select and use appropriate units and tools to measure to the degree of accuracy required in particular measurement situations.

2.3.11.B. Measure and compare angles in degrees and radians.

2.3.11.C. Demonstrate the ability to produce measures with specified levels of precision.

PA.2.4.11. Mathematical Reasoning and Connections: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.4.11.A. Use direct proofs, indirect proofs or proof by contradiction to validate conjectures.

2.4.11.B. Construct valid arguments from stated facts.

2.4.11.C. Determine the validity of an argument.

2.4.11.D. Use truth tables to reveal the logic of mathematical statements.

2.4.11.E. Demonstrate mathematical solutions to problems (e.g., in the physical sciences).

PA.2.5.11. Mathematical Problem Solving and Communication: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.5.11.A. Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.

2.5.11.B. Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results.

2.5.11.C. Present mathematical procedures and results clearly, systematically, succinctly and correctly.

2.5.11.D. Conclude a solution process with a summary of results and evaluate the degree to which the results obtained represent an acceptable response to the initial problem and why the reasoning is valid.

PA.2.6.11. Statistics and Data Analysis: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.6.11.A. Design and conduct an experiment using random sampling. Describe the data as an example of a distribution using statistical measures of center and spread. Organize and represent the results with graphs. (Use standard deviation, variance and t-tests.)

2.6.11.B. Use appropriate technology to organize and analyze data taken from the local community.

2.6.11.C. Determine the regression equation of best fit (e.g., linear, quadratic, exponential).

2.6.11.D. Make predictions using interpolation, extrapolation, regression and estimation using technology to verify them.

2.6.11.E. Determine the validity of the sampling method described in a given study.

2.6.11.F. Determine the degree of dependence of two quantities specified by a two-way table.

2.6.11.G. Describe questions of experimental design, control groups, treatment groups, cluster sampling and reliability.

2.6.11.H. Use sampling techniques to draw inferences about large populations.

2.6.11.I. Describe the normal curve and use its properties to answer questions about sets of data that are assumed to be normally distributed.

PA.2.7.11. Probability and Predictions: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.7.11.A. Compare odds and probability.

2.7.11.B. Apply probability and statistics to perform an experiment involving a sample and generalize its results to the entire population.

2.7.11.C. Draw and justify a conclusion regarding the validity of a probability or statistical argument.

2.7.11.D. Use experimental and theoretical probability distributions to make judgments about the likelihood of various outcomes in uncertain situations.

2.7.11.E. Solve problems involving independent simple and compound events.

PA.2.8.11. Algebra and Functions: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.8.11.A. Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically.

2.8.11.B. Give examples of patterns that occur in data from other disciplines.

2.8.11.C. Use patterns, sequences and series to solve routine and non-routine problems.

2.8.11.D. Formulate expressions, equations, inequalities, systems of equations, systems of inequalities and matrices to model routine and non-routine problem situations.

2.8.11.E. Use equations to represent curves (e.g., lines, circles, ellipses, parabolas, hyperbolas).

2.8.11.F. Identify whether systems of equations and inequalities are consistent or inconsistent.

2.8.11.G. Analyze and explain systems of equations, systems of inequalities and matrices.

2.8.11.H. Select and use an appropriate strategy to solve systems of equations and inequalities using graphing calculators, symbol manipulators, spreadsheets and other software.

2.8.11.I. Use matrices to organize and manipulate data, including matrix addition, subtraction, multiplication and scalar multiplication.

2.8.11.J. Demonstrate the connection between algebraic equations and inequalities and the geometry of relations in the coordinate plane.

2.8.11.K. Select, justify and apply an appropriate technique to graph a linear function in two variables, including slope-intercept, x- and y-intercepts, graphing by transformations and the use of a graphing calculator.

2.8.11.L. Write the equation of a line when given the graph of the line, two points on the line, or the slope of the line and a point on the line.

2.8.11.M. Given a set of data points, write an equation for a line of best fit.

2.8.11.N. Solve linear, quadratic and exponential equations both symbolically and graphically.

2.8.11.O. Determine the domain and range of a relation, given a graph or set of ordered pairs.

2.8.11.P. Analyze a relation to determine whether a direct or inverse variation exists and represent it algebraically and graphically.

2.8.11.Q. Represent functional relationships in tables, charts and graphs.

2.8.11.R. Create and interpret functional models.

2.8.11.S. Analyze properties and relationships of functions (e.g., linear, polynomial, rational, trigonometric, exponential, logarithmic).

2.8.11.T. Analyze and categorize functions by their characteristics.

PA.2.9.11. Geometry: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.9.11.A. Construct geometric figures using dynamic geometry tools (e.g., Geometer's Sketchpad, Cabri Geometre).

2.9.11.B. Prove that two triangles or two polygons are congruent or similar using algebraic, coordinate and deductive proofs.

2.9.11.C. Identify and prove the properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles and diagonals using deductive proofs.

2.9.11.D. Identify corresponding parts in congruent triangles to solve problems.

2.9.11.E. Solve problems involving inscribed and circumscribed polygons.

2.9.11.F. Use the properties of angles, arcs, chords, tangents and secants to solve problems involving circles.

2.9.11.G. Solve problems using analytic geometry.

2.9.11.H. Construct a geometric figure and its image using various transformations.

2.9.11.I. Model situations geometrically to formulate and solve problems.

2.9.11.J. Analyze figures in terms of the kinds of symmetries they have.

PA.2.10.11. Trigonometry: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.10.11.A. Use graphing calculators to display periodic and circular functions; describe properties of the graphs.

2.10.11.B. Identify, create and solve practical problems involving right triangles using the trigonometric functions and the Pythagorean Theorem.

PA.2.11.11. Concepts of Calculus: Pennsylvania's public schools shall teach, challenge and support every student to realize his or her maximum potential and to acquire the knowledge and skills needed to:

2.11.11.A. Determine maximum and minimum values of a function over a specified interval.

2.11.11.B. Interpret maximum and minimum values in problem situations.

2.11.11.C. Graph and interpret rates of growth/decay.

2.11.11.D. Determine sums of finite sequences of numbers and infinite geometric series.

2.11.11.E. Estimate areas under curves using sequences of areas.