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Contents
Introduction..............................................................................................xiii
1: Algebra...................................................................................................1
1.1: Variables and Equations...................................................................2
Variables, Consecutive Integers, d=rt, Harmonic
Mean, Moles Digging Holes.
1.2: Linear Equations.............................................................................10
Slope, Intercepts, Slope-Intercept Form, PointSlope Form, Standard Form, Parallel/Perpendicular Lines, Horizontal and Vertical Lines.
1.3: Systems of Equations.....................................................................16
Substitution, Elimination, Common Word
Problems, Cheating the System.
1.4: Ratios, Proportions, and Percents..................................................23
Ratios, Proportions, Percents, and Percent of
Change.
Review 1.1 to 1.4......................................................................................30
1.5: Distribution.......................................................................................31
The Distributive Property, Difference of Squares
and Perfect Squares, Sum of the Reciprocals.
1.6: Quadratics.......................................................................................35
Perfect Squares, Difference of Squares, Factoring
By Grouping, Solving Quadratics by Factoring,
The Quadratic Formula, Word Problems, Sum and
Product of the Roots.
1.7: Exponents and Radicals.................................................................44
Multiplying, Dividing, Raising a Power to a
Power, Fractional Exponents, Square Roots.
1.8: Statistics..........................................................................................49
Mean, Median, Mode and Range.
1.9: Sequences and Series.....................................................................52
Arithmetic Sequences and Series,
Geometric Sequences and Series.
1.10: More Substitution.........................................................................58
Nested Square Roots, Continued Fractions.
Chapter Review........................................................................................60
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2: Counting..............................................................................................63
2.1: Basics................................................................................................64
Simple Counting Techniques.
2.2: Venn Diagrams.................................................................................67
Two and Three-Set Venn Diagrams.
2.3: Bowling Pins and Handshakes......................................................70
Triangular Number Patterns.
2.4: Counting Outcomes.........................................................................73
The Fundamental Counting Principle.
2.5: Casework..........................................................................................75
Keeping Organized.
Review 2.1 to 2.5......................................................................................77
2.6: Factorials and Permutations...........................................................78
Factorials, Permutations, Permutations with
Restrictions.
2.7: Combinations...................................................................................84
Combinations, Paths on a Grid, Beyond Casework, Complementary Counting.
Review 2.1 to 2.7......................................................................................91
2.8: Sticks and Stones...........................................................................93
With and Without Restrictions.
2.9: Pascal’s Triangle...............................................................................97
Combinations and Pascal’s Triangle.
Chapter Review........................................................................................100
3: Probability........................................................................................103
3.1: Probability Basics..........................................................................104
Compound Events, Dependent and Independent,
Replacement, Cards.
3.2: Compound Events and Counting...............................................111
Combining Counting Techniques and Compound
Probability.
3.3: Casework and Probability.............................................................114
3.4: Probability and Combinations....................................................116
3.5: Complementary Counting and Probability................................118
Review 3.1 to 3.5......................................................................................120
3.6: Geometric Probability....................................................................121
3.7: Expected Value..............................................................................124
Finding the average outcome.
Chapter Review........................................................................................126
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Table of Contents
4: Number Theory.........................................................................129
4.1: Primes and Divisibility..................................................................130
Sieve of Eratosthenes, Primes, Divisibility Rules.
4.2: Factors............................................................................................137
The Fundamental Theorem of Arithmetic, Factor
Counting.
4.3: Factor Tricks..................................................................................143
Product of the Factors, Sum of the Factors, GCF
and LCM, Factorials.
Review 2.1 to 2.5......................................................................................154
4.4: Different Bases.............................................................................155
Notation, Basic Arithmetic.
4.5: The Units Digit.............................................................................159
Addition, Multiplication, Perfect Squares,
Exponents.
4.6: Fractions and Decimals................................................................163
Repeating/Terminating, Conversions.
4.7: Modular Arithmetic......................................................................167
Chapter Review........................................................................................172
5: Geometry......................................................................................175
5.1: Geometry Basics...........................................................................176
Lines, Angles, Triangles, Polygons, Trapezoids,
Parallelograms, Kites.
5.2: Circles.............................................................................................185
Chords, Secants, Tangents, Arcs, Inscribed
Angles, Properties, Circumference and Arc
Length.
5.3: Pythagorean Theorem..................................................................193
Triples, Special Right Triangles, Distance
Formula, Algebra and the Pythagorean Theorem.
5.4: Area.................................................................................................204
Formulas, Heron’s Formula, Altitudes, Working
Backwards.
5.5: Three-Dimensional Geometry......................................................208
Polyhedra, Faces, Vertices, and Edges, Volume.
5.6: Similarity.........................................................................................215
Right Triangles, Circles, Parallel Lines, Area and
Volume Relationships.
Chapter Review........................................................................................226
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6: Solutions...........................................................................................229
Algebra
Key..........................................................................................230
Solutions................................................................................232
Counting
Key..........................................................................................260
Solutions................................................................................261
Probability
Key..........................................................................................288
Solutions................................................................................289
Number Theory
Key..........................................................................................308
Solutions................................................................................310
Geometry
Key..........................................................................................334
Solutions................................................................................336
7: Appendix.............................................................................................363
Useful Lists and Formulas.
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