Major: Math
"Math Major! Ito ang specialization para sa mga gustong maging secondary Math teacher. Algebra, Geometry, Trigonometry, Statistics, at Calculus - lahat 'to covered! Problem solving is your superpower. Ready ka na ba mag-find ng X at Y? Let's do some Math, future Math teacher!"
1. Algebra 📐
Linear Equations & Inequalities
Solving Linear Equations
ax + b = c → x = (c - b)/a
Example: 2x + 5 = 11
2x = 6 → x = 3
Slope-Intercept Form
y = mx + b
m = slope (rise/run)
b = y-intercept
🔢 Systems of Equations
Substitution
Solve for one variable, substitute into other equation
Elimination
Add/subtract equations to eliminate a variable
Graphing
Find intersection point of two lines
📊 Quadratic Equations
Standard Form: ax² + bx + c = 0
Quadratic Formula:
x = (-b ± √(b²-4ac)) / 2a
Discriminant (b²-4ac):
- > 0: Two real solutions
- = 0: One real solution
- < 0: No real solutions
2. Geometry 📐
Area & Perimeter Formulas
| Shape | Area | Perimeter/Circumference |
|---|---|---|
| Rectangle | A = l × w | P = 2(l + w) |
| Square | A = s² | P = 4s |
| Triangle | A = ½bh | P = a + b + c |
| Circle | A = πr² | C = 2πr = πd |
| Trapezoid | A = ½(b₁ + b₂)h | P = sum of all sides |
🔺 Pythagorean Theorem
For right triangles:
a² + b² = c²
c = hypotenuse (longest side, opposite 90°)
📏 Distance Formula
d = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
🎲 Volume Formulas
Cube
V = s³
Rectangular Prism
V = lwh
Cylinder
V = πr²h
Sphere
V = (4/3)πr³
3. Trigonometry 📐
SOH-CAH-TOA (Right Triangle)
SOH
sin θ = Opposite / Hypotenuse
CAH
cos θ = Adjacent / Hypotenuse
TOA
tan θ = Opposite / Adjacent
📊 Special Angles
| θ | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
🔄 Identities
- Pythagorean: sin²θ + cos²θ = 1
- Reciprocals:
csc = 1/sin, sec = 1/cos, cot = 1/tan - Quotient: tan = sin/cos
🔺 Law of Sines & Cosines
Law of Sines:
a/sin A = b/sin B = c/sin C
Law of Cosines:
c² = a² + b² - 2ab·cos C
4. Statistics & Probability 📊
Measures of Central Tendency
Mean (Average)
Sum of all values ÷ Number of values
x̄ = Σx/n
Median
Middle value when arranged in order
If even # of values, average middle two
Mode
Most frequently occurring value
Can have multiple modes or no mode
📈 Measures of Dispersion
- Range: Max - Min
- Variance (σ²): Average of squared deviations
- Standard Deviation (σ): √Variance
- Coefficient of Variation: (σ/x̄) × 100%
🎲 Basic Probability
- P(event) = Favorable outcomes / Total outcomes
- 0 ≤ P ≤ 1 (0 = impossible, 1 = certain)
- P(A') = 1 - P(A) [complement]
- P(A or B) = P(A) + P(B) - P(A and B)
🔢 Permutations & Combinations
Permutation (order matters)
P(n,r) = n! / (n-r)!
Example: Arranging books on shelf
Combination (order doesn't matter)
C(n,r) = n! / [r!(n-r)!]
Example: Choosing committee members
5. Calculus ∫
Differential Calculus (Derivatives)
What is a Derivative?
Rate of change / Slope of tangent line
f'(x) = lim[h→0] (f(x+h) - f(x))/h
Basic Rules
- Power Rule: d/dx(xⁿ) = nxⁿ⁻¹
- Constant: d/dx(c) = 0
- Sum Rule: d/dx(f+g) = f' + g'
Integral Calculus (Antiderivatives)
What is an Integral?
Area under the curve / Antiderivative
∫ = "reverse" of derivative
Basic Rules
- Power Rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C
- Constant: ∫cdx = cx + C
- Don't forget +C! (constant of integration)
📍 Limits
Foundation of calculus - what value a function approaches
Direct Substitution: Plug in the value
Factoring: If you get 0/0
lim[x→∞] 1/x = 0
L'Hôpital's Rule: For indeterminate forms
6. Problem-Solving Strategies 🧠
George Polya's 4 Steps
1. Understand
What is asked? What is given?
2. Plan
What strategy will you use?
3. Execute
Carry out the plan
4. Look Back
Check and verify answer
🔄 Work Backwards
Start from the answer and reverse the steps. Good for "find the original" problems.
✏️ Draw a Diagram
Visualize the problem. Essential for geometry and word problems.
🔢 Look for Patterns
Find sequences, relationships, or regularities in the data.
🧩 Simplify
Break complex problems into smaller, manageable parts.
7. Math Teaching Methodologies 👩🏫
| Approach | Description | Best For |
|---|---|---|
| CPA (Concrete-Pictorial-Abstract) | Manipulatives → Drawings → Symbols | Concept introduction |
| Problem-Based Learning | Start with real-world problems | Application, critical thinking |
| Cooperative Learning | Group work, peer learning | Communication, collaboration |
| Discovery/Inquiry | Students discover concepts themselves | Deep understanding |
🎯 K-12 Math Competencies
📋 Practice Questions
1. Solve for x: 3x - 7 = 14
Show Answer
3x = 21 → x = 7
2. Find the area of a circle with radius 5 cm.
Show Answer
A = πr² = π(5)² = 25π ≈ 78.54 cm²
3. In a right triangle, if sin θ = 3/5, find cos θ.
Show Answer
Using Pythagorean identity: sin²θ + cos²θ = 1
(3/5)² + cos²θ = 1
cos²θ = 1 - 9/25 = 16/25
cos θ = 4/5
4. Find the mean of: 5, 8, 12, 15, 20
Show Answer
Mean = (5+8+12+15+20)/5 = 60/5 = 12
5. Find the derivative of f(x) = 3x² + 2x - 5
Show Answer
f'(x) = 6x + 2 (using power rule: bring down exponent, reduce by 1)
🎯 Exam Strategy Tips
📐 Algebra
Master solving linear and quadratic equations. Know the quadratic formula by heart. Practice systems of equations.
📏 Geometry
Memorize area and volume formulas. Pythagorean theorem is ESSENTIAL. Know distance and midpoint formulas.
📐 Trigonometry
SOH-CAH-TOA is your best friend. Memorize special angles (30°, 45°, 60°). Know Law of Sines and Cosines.
📊 Statistics
Know mean, median, mode, and standard deviation. Understand probability basics and counting principles.
∫ Calculus
Understand concept of limits, derivatives (rate of change), integrals (area). Know basic differentiation rules.
🧠 Problem Solving
Follow Polya's 4 steps: Understand → Plan → Execute → Look Back. Show your solutions clearly!
Math Major = Practice, Practice, Practice! Solve problems daily! 📐✏️
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