High School Pre-Calculus

Student Learnings: What students should know and be able to do

Mathematical Reasoning

Apply skills of mathematical representation, communication and reasoning throughout the remaining three content strands.

• Assess the reasonableness of a solution by comparing the solution to appropriate graphical or numerical estimates or by recognizing the feasibility of solutions in a given context and rejecting extraneous solutions.
• Appropriately use examples and counterexamples to make and test conjectures, justify solutions, and explain results.
• Translate a problem described verbally or by tables, diagrams or graphs, into suitable mathematical language, solve the problem mathematically and interpret the result in the original context.
• Support mathematical results by explaining why the steps in a solution are valid and why a particular solution method is appropriate.
• Determine whether or not relevant information is missing from a problem and if so, decide how to best express the results that can be obtained without that information.
• Know and use the relationship that exists among a logical implication of the form “if A, then B,” its converse “if B, then A,” its inverse “if not A, then not B,” and its contrapositive “if not B, then not A.”

Number Sense, Computation and Operations

Number Sense
Use real numbers, represented in a variety of ways, to quantify information and to solve real-world and mathematical problems.

• Use real numbers, represented in a variety of ways, to quantify information and to solve real-world and mathematical problems.

Computation and Operation
Appropriately use calculators and other technologies to solve algebraic, geometric, probabilistic and statistical problems.

• Apply the correct order of operations and grouping symbols when using calculators and other technologies.
• Know, use and translate calculator notational conventions to mathematical notation.
• Recognize the impact of units such as degrees and radians on calculations.
• Recognize that applying an inverse function with a calculator may lead to extraneous or incomplete solutions.
• Understand the limitations of calculators such as missing or additional features on graphs due to viewing parameters or misleading representations of zero or very large numbers
• Understand that use of a calculator requires appropriate mathematical reasoning and does not replace the need for mental computation.

Patterns, Functions, and Algebra

Patterns and Functions
Represent and analyze real-world and mathematical problems using numeric, graphic and symbolic methods for a variety of functions.

• Know the numeric, graphic and symbolic properties of linear, step, absolute value and quadratic functions. Graphic properties may include rates of change, intercepts, maxima and minima.
• Model exponential growth and decay, numerically, graphically and symbolically, using exponential functions with integer inputs.
• Analyze the effects of coefficient changes on linear and quadratic functions and their graphs.
• Apply basic concepts of linear, quadratic and exponential expressions or equations in real-world problems such as loans, investments and the path of a projectile.
• Distinguish functions from other relations using graphic and symbolic methods.

Algebra (Algebraic Thinking)
Solve simple equations and inequalities numerically, graphically and symbolically. Use recursion to model and solve real-world and mathematical problems.

• Translate among equivalent forms of expressions, such as, simplify algebraic expressions involving nested pairs of parentheses and brackets, simplify rational expressions, factor a common term from an expression and apply associative, commutative and distributive laws.
• Understand the relationship between absolute value and distance on the number line and graph simple expressions involving absolute value such as, [x-3] = 6 or [x÷2] < 5.
• Find equations of a line given two points on the line, a point and the slope of the line or the slope and the y-intercept of the line.
• Translate among equivalent forms of linear equations and inequalities.
• Use a variety of models such as equations, inequalities, algebraic formulas, written statements, tables and graphs or spreadsheets to represent functions and patterns in real-world and mathematical problems.
• Apply the laws of exponents to perform operations on expressions with integer exponents.
• Solve linear equations and inequalities in one variable with numeric, graphic and symbolic methods.
• Find real solutions to quadratic equations in one variable with numeric, graphic and symbolic methods.
• Use appropriate terminology and mathematical notation to define and represent recursion.
• Create and use recursive formulas to model and solve real-world and mathematical problems.
• Solve systems of two linear equations and inequalities with two variables using numeric, graphic and symbolic methods.
• Understand how slopes can be used to determine whether lines are parallel or perpendicular. Given a line and a point not on the line, find the equations for the lines passing through that point and parallel or perpendicular to the given line.

Data Analysis, Statistics and Probability

Probability
Use appropriate counting procedures, calculate probabilities in various ways and apply theoretical probability concepts to solve real-world and mathematical problems.

• Select and apply appropriate counting procedures to solve real-world and mathematical problems, including probability problems.
• Use probability models, including area and binomial models, in real-world and mathematical problems.

Spatial Sense, Geometry and Measurement

Geometry
Apply basic theorems of plane geometry, right triangle trigonometry, coordinate geometry and a variety of visualization tools to solve real-world and mathematical problems.

• Know and use theorems about triangles and parallel lines in elementary geometry to justify facts about various geometrical figures and solve real-world and mathematical problems. These theorems include criteria for two triangles to be congruent or similar and facts about parallel lines cut by a transversal.
• Know and use properties of two- and three-dimensional figures to solve real-world and mathematical problems such as finding area, perimeter, volume and surface area; applying direct or indirect methods of measurement; the Pythagorean theorem and its converse; and properties of 45º-45º-90º triangles.
• Apply the basic concepts of right triangle trigonometry including sine, cosine and tangent to solve real-world and mathematical problems.
• Use coordinate geometry to represent and examine geometric concepts such as the distance between two points, the midpoint of a line segment, the slope of a line and the slopes of parallel and perpendicular lines.
• Use numeric, graphic and symbolic representations of transformations such as reflections, translations and change of scale in one-, two- and three-dimensions to solve real-world and mathematical problems.

Measurement
Use the interconnectedness of geometry, algebra and measurement to explore real-world mathematical problems.

• Use the interconnectedness of geometry, algebra and measurement to explore real-world mathematical problems.

Algebra

• Demonstrate facility with a wide range of algebraic operations and use the relationship between coordinate geometry and algebraic equations to solve real-world and mathematical problems.
• Perform the four arithmetic operations with polynomials, except that division is restricted to division by monomials and linear binomials.
• Simplify a wide variety of algebraic expressions, including those in which numerator or denominator needs to be rationalized.
• Apply the laws of exponents to perform operations on expressions with fractional exponents.
• Know the numeric, graphic and symbolic properties of power, logarithmic and exponential functions.
• Solve a wide variety of mathematical and real-world problems involving power, exponential and logarithmic functions and equations, discard extraneous solutions and present results graphically.
• Know the numeric, graphic and symbolic properties of rational functions.
• Solve a wide variety of mathematical and real-world problems involving rational function, discard extraneous solutions and present results graphically.
• Factor polynomials representing the difference of squares, perfect square trinomials and quadratics with rational factors.
• Add, subtract, multiply and divide complex numbers, interpret sums geometrically, and find complex solutions of quadratic equations.
• Know and use the Factor and Remainder Theorems.
• Find the inverse of a function and the composition of functions by numeric and symbolic methods. Know the relationship between the graphs of a function and its inverse.
• Know and use formal notation for sequences and series to solve related problems.

Trigonometry and Geometry

Understand the properties of the standard trigonometric functions and apply them to real-world and mathematical problems, especially geometrical problems. Develop increased mastery of geometric proof methodology.

• Know the six trigonometric functions defined for an angle in a right triangle.
• Given the coordinates of a point on the terminal side of an angle in standard position in the xy-plane, find the values of the trigonometric functions.
• Convert between degrees and radian measures.
• Solve applied problems about triangles using the law of sines including the ambiguous case.
• Solve applied problems about triangles using the law of cosines.
• Graph the functions of the form Asin(Bt+C), Acos(Bt+C), and Atan(Bt+C) and know the meaning of terms frequency, amplitude, phase shift and period.
• Simplify trigonometric expressions using identifies and verify simple trigonometric identities including sin²x + cos²x = 1, sum difference, double angle and half-angle formulas for sine and cosine.
• Find all the solutions of a trigonometric equation on various intervals.
• Know and be able to use the definitions of the inverse trigonometric functions and related methods to solve problems such as find cos(x) and tan(x) given the value of sin x and the quadrant containing the terminal side.

Copyright 2004, Pearson/Prentice Hall
Pre-Calculus: Graphical, Numerical, Algebraic, Demana, et al