I. ARITHMETIC

A. Whole numbers

1. Operations—addition, subtraction, multiplication, division

2. Prime and composite numbers

3. Factors and divisors

B. Fractions

1. Types—proper, improper, mixed numbers

2. Operations

C. Decimals

1. Operations

2. Conversions

a) Decimals to fractions

b) Fractions to decimals

3. Rounding and approximation

4. Powers of 10

a) Multiplication

b) Division

c) Scientific notation

D. Percent

1. Conversions

a) Percent to decimal

b) Decimal to percent

2. Percent problems

E . Ratio and proportion

F . Square roots

G. Averages

H . Metric measurement

 

 

II. ALGEBRA

A . Signed numbers

1. Absolute value

2. Inequality and order of signed numbers

3. Addition, subtraction, multiplication, division

4. Order of operations

5. Grouping symbols

6. Evaluating algebraic expressions and formulas

B. Properties of operations

1. Commutative properties

2. Associative properties

3. Distributive properties

4. Special properties of zero

5. Special properties of one

6. Additive and multiplicative inverses

C . Operations with polynomials

1. Exponents and coefficients

2. Addition and subtraction

3. Multiplication

4. Division

D . Equations in one variable

1. Methods of solution

2. Literal equations

E . Inequalities in one variable

F . Systems of equations and inequalities in two variables

G. Verbal Problems

1. Number

2. Consecutive integer

3. Motion

4. Coin

5. Mixture

6. Age

7. Work

8. Variation—direct and inverse

H. Special products and factoring

1. Common monomial factors

2. Trinomials of the form ax2 + bx + c

3. Difference of two squares

4. Complete factoring

I. Algebraic fractions

1. Simplifying fractions

2. Multiplication

3. Division

4. Addition and subtraction

a) Same denominators

b) Different denominators

5. Complex fractions

6. Equations involving fractions

J . Radicals and irrational numbers

1. Simplifying radicals

2. Addition and subtraction of radicals

3. Multiplication and division of radicals

4. Rationalizing denominators

5. Radical equations

6. Fractional exponents

K. Solution of quadratic equations

1. Factoring

2. Completing the square

3. Formula

L. Graphing

1. Ordered pairs in the plane

2. Methods of graphing linear equations

a) Pairs in the solution set

b) Intercepts

c) Slope and slope-intercept method

3. Parallel and perpendicular lines

4. Graphing inequalities

5. Graphical solution of systems of equations

M . Solution of simple cubic equations

1. Factor theorem

2. Remainder theorem

3. Synthetic division

4. Irrational and complex roots

5. Solving simple cubic equations

 

 

III. GEOMETRY

A . Angles

1. Types—acute, right, obtuse

2. Complements and supplements

3. Vertical angles

B . Lines

1. Parallel lines and their angles

2. Perpendicular lines

C. Triangles

1. Sum of the angles

2. Congruent triangles

3. Similar triangles

4. Special triangles

a) Isosceles

b) Equilateral

c) Right (Pythagorean Theorem)

5. Vectors

D . Polygons

1. Quadrilaterals

a) Parallelogram

b) Rectangle

c) Square

d) Rhombus

e) Trapezoid

f) Regular Polygons

E. Circles

1. Special lines and their related angles

a) Radius and diameter

b) Chord

c) Tangent

d) Secant

2. Angle and arc measurement

3. Polygons inscribed in circles

F . Perimeter and area

1. Triangles

2. Polygons

3. Circles

a) Circumference and arc length

b) Area of sectors and segments

G . Volume

1. Pyramid

2. Prism

3. Cylinder

4. Cone

5. Sphere

6. Cube

7. Rectangular solid

H . Coordinate geometry

1. Coordinate representation of points

2. Distance between two points

3. Midpoint of a line segment

4. Slope of a line

5. Parallel and perpendicular lines

I. Basic trigonometry

1. Definitions of sine, cosine, tangent

2. Trigonometry in special triangles

a) 30°–60°–90° triangle

b) Isoceles right triangle

3. Trigonometric problems

a) Angle of elevation

b) Angle of depression

 

 

 

IV. FUNCTIONS AND THEIR GRAPHS

A . Relations and functions

1. Ordered pairs

2. Function notation

3. Domain and range

4. One-to-one functions

5. Inverse functions

6. Combining functions

a) Addition, subtraction, multiplication, division

b) Composition

B. Graphs

1. Linear

a) Slope

b) Intercepts

2. Special functions

a) Absolute value function

b) Step functions

3. Polynominal and rational functions

a) Quadratic—parabola

i. Axis of symmetry

ii. Vertex

b) Cubics

c) Hyperbola of the form xy = k

4. Related non-function graphs

a) Circle

b) Ellipse

c) Hyperbola of the form ax2 – by2 = c

5. Graphs of inverse functions

 

 

 

V. REAL NUMBER SYSTEM

A . Subsets of the real numbers

1. Natural numbers

a) Primes

b) Composites—prime factorization

2. Integers

a) Multiples and divisors

i. Factors

ii. Divisibility

iii. Least common multiple

iv. Greatest common divisor

v. Perfect squares

b) Odd and even integers

3. Rational and irrational numbers

a) Decimal representations

b) Simplification of radicals and exponents

c) Identifying rational and irrational numbers

B . Operations and properties

1. Properties of the binary operations

a) Closure

b) Commutative properties

c) Associative properties

d) Distributive properties

2. Absolute value

3. Real number line

a) Order

b) Density

c) Completeness

4. Properties of zero and one

a) Identity elements

b) Additive and multiplicative inverses

c) Division involving zero

d) Zero as an exponent

5. Nature of the roots of quadratic equations

6. Pythagorean triples

 

 

VI. LOGIC

A . Propositions

1. Simple statements

a) Symbols

b) Quantifiers (all, some)

2. Negation

3. Compound statements

a) Conjunction

b) Disjunction

c) Implication (conditional statements)

i. Necessary conditions

ii. Sufficient conditions

iii. Equivalence (necessary and sufficient conditions)

d) Derived implications

i. Converse

ii. Inverse

iii. Contrapositive

B . Truth tables

C . Methods of proof

1. Valid arguments

a) Direct

b) Indirect—contradiction and counterexample

2. Invalid arguments—fallacies

 

 

VII. SETS

A . Meaning and symbols

1. Set notation

2. Set membership

3. Ordered pairs

4. Cardinality of a set

B . Types of sets

1. Finite

2. Infinite

3. Empty

C. Relationships between sets

1. Equal sets

2. Equivalent sets

3. Subsets

4. Complements

D. Set Operations

1. Union

2. Intersection

3. Cartesian products

4. Laws of set operations

5. Closure

E . Venn diagrams

 

 

 

 

VIII. TRIGONOMETRY

A. Trigonometry of the right triangle

1. Definitions of the six functions

2. Relations of the functions of the complementary angles

3. Reciprocal relations among the functions

4. Variations in the functions of acute angles

5. Pythagorean and quotient relations

6. Functions of 30°, 45°, and 60°

7. Applications of the functions to right triangle problems

B. Trigonometric functions of the general angle

1. Generating an angle of any size

2. Radians and degrees

3. Using radians to determine arc length

4. Definitions of the functions of an angle

5. Signs of the functions in the four quadrants

6. Functions of the quadrantal angle

7. Finding the value of functions of any angle

C . Identities and equations

1. Difference between identities in equations

2. Proving identities

3. Solving linear trigonometric functions

4. Solving trigonometric quadratic equations

D . Generalized trigonometric relationships

1. Functions of the sum of two angles

2. Functions of the difference of two angles

3. Functions of the double angle

4. Functions of the half angle

E . Graphs of trigonometric functions

1. Graphs of the sine, cosine, and tangent curves

2. Properties of the sine, cosine, and tangent curves

3. Definitions of amplitude, period, and frequency

4. Solving trigonometric equations graphically

F . Solutions of oblique triangles

1. Law of sines

2. Law of cosines

3. Using logarithms to solve oblique triangle problems

4. Vector problems—parallelogram of forces

5. Navigation problems

 

 

 

 

 

IX. MISCELLANEOUS TOPICS

A. Complex numbers

1. Meaning

2. Operations

a) Addition and subtraction

b) Multiplication and division

i. Powers of i

ii. Complex conjugate

3. Complex roots of quadratic equations

B . Number Bases

1. Converting from base 10 to other bases

2. Converting from other bases to base 10

3. Operations in other bases

C . Exponents and logarithms

1. Meaning of logarithms

2. Computation with exponents and logarithms

3. Equations

4. Graphs of exponential and logarithmic functions

D . Binary operations

1. Definition of binary operations

2. Properties of binary operations

3. Application to modular arithmetic

E . Identity and inverse elements

1. Addition

2. Multiplication

3. Other operations