Lesson 250 min read

Mathematics

Algebra, Geometry, Statistics, and Problem Solving

In This Lesson

1. Algebra Fundamentals 2. Geometry Essentials 3. Statistics & Probability 4. Number Theory 5. Word Problems

⚠️ No Calculator Allowed

DLSUCET does not allow calculators. Practice mental math and learn shortcuts for faster computation. Master estimation techniques for checking answers.

Algebra Fundamentals

Exponent Rules

a^m × aⁿ = a^m+n (Product rule)
a^m ÷ aⁿ = a^m-n (Quotient rule)
(a^m)ⁿ = a^mn (Power rule)
(ab)ⁿ = aⁿ × bⁿ (Product to power)
a⁰ = 1 (Zero exponent)
a^-n = 1/aⁿ (Negative exponent)

Radical Rules

√(ab) = √a × √b
√(a/b) = √a / √b
√a² = |a|
a^(1/n) = ⁿ√a
Rationalizing: 1/√a = √a/a

Factoring Patterns

Difference of squares: a² - b² = (a+b)(a-b)
Perfect square trinomial: a² ± 2ab + b² = (a ± b)²
Sum of cubes: a³ + b³ = (a+b)(a² - ab + b²)
Difference of cubes: a³ - b³ = (a-b)(a² + ab + b²)

Quadratic Formula

For ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

Discriminant (b² - 4ac):

> 0: Two real solutions
= 0: One real solution
< 0: No real solutions

Linear Equations

Slope-intercept: y = mx + b
Point-slope: y - y₁ = m(x - x₁)
Slope formula: m = (y₂ - y₁)/(x₂ - x₁)
Parallel lines: Same slope (m₁ = m₂)
Perpendicular: m₁ × m₂ = -1

Geometry Essentials

Area Formulas

Rectangle: A = l × w

Triangle: A = ½bh

Circle: A = πr²

Trapezoid: A = ½(b₁ + b₂)h

Parallelogram: A = bh

Rhombus: A = ½d₁d₂

Volume Formulas

Cube: V = s³

Rectangular prism: V = lwh

Cylinder: V = πr²h

Cone: V = ⅓πr²h

Sphere: V = ⁴⁄₃πr³

Pyramid: V = ⅓Bh

Pythagorean Theorem

For right triangles: a² + b² = c²

Common Pythagorean Triples:

3, 4, 5 (and multiples: 6,8,10 | 9,12,15)
5, 12, 13
8, 15, 17
7, 24, 25

Special Right Triangles

45-45-90 Triangle

Sides: 1 : 1 : √2

Leg : Leg : Hypotenuse

30-60-90 Triangle

Sides: 1 : √3 : 2

Short : Long : Hypotenuse

Circle Properties

Circumference: C = 2πr = πd
Arc length: L = (θ/360°) × 2πr
Central angle: Equals intercepted arc
Inscribed angle: Half of intercepted arc
Tangent: Perpendicular to radius at point of tangency

Statistics & Probability

Measures of Central Tendency

Mean (Average): Sum of all values ÷ Number of values
Median: Middle value when arranged in order
Mode: Most frequently occurring value

Tip: For even number of values, median = average of two middle values

Measures of Spread

Range: Maximum - Minimum
Variance: Average of squared deviations
Standard Deviation: √Variance
Interquartile Range: Q3 - Q1

Basic Probability

P(event) = Favorable outcomes / Total outcomes

0 ≤ P(event) ≤ 1
P(not A) = 1 - P(A)
OR (union): P(A or B) = P(A) + P(B) - P(A and B)
AND (independent): P(A and B) = P(A) × P(B)

Counting Principles

Multiplication: If A can happen in m ways and B in n ways, then A and B can happen in m × n ways
Permutation (order matters): P(n,r) = n!/(n-r)!
Combination (order doesn't matter): C(n,r) = n!/[r!(n-r)!]

Common Probability Scenarios

Coin flip: P(heads) = 1/2
Die roll: P(any number) = 1/6
Deck of cards: P(specific card) = 1/52
Two dice: P(sum of 7) = 6/36 = 1/6

Number Theory

Divisibility Rules

By 2: Last digit is even
By 3: Sum of digits divisible by 3
By 4: Last two digits divisible by 4
By 5: Ends in 0 or 5
By 6: Divisible by both 2 and 3
By 9: Sum of digits divisible by 9
By 10: Ends in 0

Prime Numbers

First 20 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71

2 is the only even prime number
Prime factorization: Express number as product of primes
To check if n is prime, test divisibility up to √n

GCF and LCM

GCF: Greatest Common Factor - largest number that divides both
LCM: Least Common Multiple - smallest number divisible by both
Relationship: GCF(a,b) × LCM(a,b) = a × b

Example: GCF(12,18) = 6, LCM(12,18) = 36

Fractions, Decimals, Percents

1/2 = 0.5 = 50%
1/3 = 0.333... = 33.33%
1/4 = 0.25 = 25%
1/5 = 0.2 = 20%
1/8 = 0.125 = 12.5%
Percent change: (New - Old) / Old × 100%

Word Problems

Problem-Solving Strategy

Read carefully: Understand what's being asked
Identify: What do you know? What do you need?
Translate: Convert words to math expressions
Solve: Apply appropriate formulas
Check: Does the answer make sense?

Rate/Work Problems

Distance: D = Rate × Time
Work: Work = Rate × Time
Combined work: 1/T = 1/T₁ + 1/T₂

Example: If A takes 6 hours and B takes 3 hours, together: 1/T = 1/6 + 1/3 = 1/2, so T = 2 hours

Age Problems

Let current age = x, then:

Age in n years = x + n
Age n years ago = x - n
Twice as old = 2x

Mixture Problems

Amount × Concentration = Total substance
Total substance₁ + Total substance₂ = Final total substance
Final concentration = Total substance / Total amount

Common Word Problem Keywords

Sum, total, increased by: Addition (+)

Difference, less than: Subtraction (-)

Product, times, of: Multiplication (×)

Quotient, per, ratio: Division (÷)

Is, equals, gives: Equal sign (=)

At least: Greater than or equal (≥)

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