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# Mathematics Curriculum Framework - November 2000

## Learning Standards for Precalculus

Note: The parentheses at the end of a learning standard contain the code number for the corresponding standard in the two-year grade spans.

Number Sense and Operations

• Understand numbers, ways of representing numbers, relationships among numbers, and number systems
• Understand meanings of operations and how they relate to one another
• Compute fluently and make reasonable estimates
 Students engage in problem solving, communicating, reasoning, connecting, and representing as they: PC.N.1 Plot complex numbers using both rectangular and polar coordinates systems. Represent complex numbers using polar coordinates, i.e., a + bi = r(cosθ + isinθ). Apply DeMoivre's theorem to multiply, take roots, and raise complex numbers to a power.

Patterns, Relations, and Algebra

• Understand patterns, relations, and functions
• Represent and analyze mathematical situations and structures using algebraic symbols
• Use mathematical models to represent and understand quantitative relationships
• Analyze change in various contexts

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:

PC.P.1 Use mathematical induction to prove theorems and verify summation formulas, e.g., verify
 n k2 k=1 = n(n+1)(2n+1) 6
.
PC.P.2 Relate the number of roots of a polynomial to its degree. Solve quadratic equations with complex coefficients.
PC.P.3 Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions.
PC.P.4 Explain the identity sin2θ + cos2θ = 1. Relate the identity to the Pythagorean theorem.
PC.P.5 Demonstrate an understanding of the formulas for the sine and cosine of the sum or the difference of two angles. Relate the formulas to DeMoivre's theorem and use them to prove other trigonometric identities. Apply to the solution of problems.
PC.P.6 Understand, predict, and interpret the effects of the parameters a, ω, b, and c on the graph of y = asin(ω(x - b)) + c; similarly for the cosine and tangent. Use to model periodic processes. (12.P.13)
PC.P.7 Translate between geometric, algebraic, and parametric representations of curves. Apply to the solution of problems.
PC.P.8 Identify and discuss features of conic sections: axes, foci, asymptotes, and tangents. Convert between different algebraic representations of conic sections.
PC.P.9 Relate the slope of a tangent line at a specific point on a curve to the instantaneous rate of change. Explain the significance of a horizontal tangent line. Apply these concepts to the solution of problems.

Geometry

• Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
• Specify locations and describe spatial relationships using coordinate geometry and other representational systems
• Apply transformations and use symmetry to analyze mathematical situations
• Use visualization, spatial reasoning, and geometric modeling to solve problems
 Students engage in problem solving, communicating, reasoning, connecting, and representing as they: PC.G.1 Demonstrate an understanding of the laws of sines and cosines. Use the laws to solve for the unknown sides or angles in triangles. Determine the area of a triangle given the length of two adjacent sides and the measure of the included angle. (12.G.2) PC.G.2 Use the notion of vectors to solve problems. Describe addition of vectors, multiplication of a vector by a scalar, and the dot product of two vectors, both symbolically and geometrically. Use vector methods to obtain geometric results. (12.G.3) PC.G.3 Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems. (12.G.5)

Measurement

• Understand measurable attributes of objects and the units, systems, and processes of measurement
• Apply appropriate techniques, tools, and formulas to determine measurements
 Students engage in problem solving, communicating, reasoning, connecting, and representing as they: PC.M.1 Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular problems involving angular velocity and acceleration. (12.M.1) PC.M.2 Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. (12.M.2)

Data Analysis, Statistics, and Probability

• Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
• Select and use appropriate statistical methods to analyze data
• Develop and evaluate inferences and predictions that are based on data
• Understand and apply basic concepts of probability
 Students engage in problem solving, communicating, reasoning, connecting, and representing as they: PC.D.1 Design surveys and apply random sampling techniques to avoid bias in the data collection. (12.D.1) PC.D.2 Apply regression results and curve fitting to make predictions from data. (12.D.3) PC.D.3 Apply uniform, normal, and binomial distributions to the solutions of problems. (12.D.4) PC.D.4 Describe a set of frequency distribution data by spread (variance and standard deviation), skewness, symmetry, number of modes, or other characteristics. Use these concepts in everyday applications. (12.D.5) PC.D.5 Compare the results of simulations (e.g., ran-dom number tables, random functions, and area models) with predicted probabilities. (12.D.7)

Learning Standards for Algebra II

Appendix I: