Lesson 2
Mathematics
In This Lesson
1. Arithmetic Operations
Master the fundamental operations and order of operations (PEMDAS/GEMDAS).
Order of Operations (PEMDAS)
- Parentheses (or Grouping symbols)
- Exponents (powers and roots)
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Example
Solve: 3 + 4 × 2² - (6 ÷ 2)
- Parentheses: (6 ÷ 2) = 3
- Exponents: 2² = 4
- Multiplication: 4 × 4 = 16
- Addition/Subtraction: 3 + 16 - 3 = 16
2. Fractions & Decimals
Fractions
- Addition/Subtraction: Find LCD first
- Multiplication: Multiply numerators and denominators
- Division: Flip the divisor and multiply
- Simplifying: Divide by GCF
Decimals
- Addition/Subtraction: Align decimal points
- Multiplication: Count total decimal places
- Division: Move decimal to make divisor whole
- Conversion: Divide numerator by denominator
Common Fraction-Decimal-Percent Equivalents
3. Algebra
Algebra involves solving equations and working with variables.
Solving Linear Equations
- Simplify both sides (distribute, combine like terms)
- Move variables to one side (add/subtract)
- Move constants to the other side
- Divide by the coefficient
Example: Solve 3x + 5 = 2x + 12
3x - 2x = 12 - 5
x = 7
Factoring
- • Common factor: ax + ay = a(x + y)
- • Difference of squares: a² - b² = (a+b)(a-b)
- • Trinomial: x² + 5x + 6 = (x+2)(x+3)
Quadratic Formula
For ax² + bx + c = 0:
x = (-b ± √(b²-4ac)) / 2a
4. Geometry
Essential formulas for area, perimeter, and volume.
2D Shapes
Rectangle
Area = l × w
Perimeter = 2(l + w)
Triangle
Area = ½ × b × h
Perimeter = a + b + c
Circle
Area = πr²
Circumference = 2πr
Trapezoid
Area = ½(b₁ + b₂) × h
3D Shapes
Rectangular Prism
Volume = l × w × h
Surface Area = 2(lw + lh + wh)
Cylinder
Volume = πr²h
Surface Area = 2πr² + 2πrh
Sphere
Volume = (4/3)πr³
Surface Area = 4πr²
Cone
Volume = (1/3)πr²h
Pythagorean Theorem
For right triangles: a² + b² = c²
Where c is the hypotenuse (longest side)
5. Statistics
Measures of central tendency and data analysis.
Measures of Central Tendency
- Mean (Average): Sum of all values ÷ Number of values
Example: {2, 4, 6, 8, 10} → Mean = 30/5 = 6
- Median: Middle value when arranged in order
For even count, average the two middle values
- Mode: Most frequently occurring value
A set can have no mode, one mode, or multiple modes
Other Measures
- Range: Highest value - Lowest value
- Weighted Average: Σ(value × weight) ÷ Σweights
- Probability: Favorable outcomes ÷ Total possible outcomes
6. Word Problems
Strategies for solving mathematical word problems.
Problem-Solving Steps
- Read the problem carefully (identify what is being asked)
- Identify the given information
- Determine what operation(s) to use
- Set up the equation
- Solve and check your answer
Common Types
- Age Problems: Use variables for ages, set up equations
- Distance Problems: Distance = Rate × Time
- Work Problems: Combined rate = 1/time₁ + 1/time₂
- Percentage Problems: Part = Whole × Percentage
- Ratio Problems: Set up proportions
Example: Age Problem
Maria is 5 years older than Juan. In 3 years, their ages will total 35. How old is Juan now?
Let x = Juan's age now
Maria's age = x + 5
In 3 years: (x + 3) + (x + 5 + 3) = 35
2x + 11 = 35 → x = 12
Juan is 12 years old.
PUPCET Math Tips
- ✓Memorize formulas - know area, perimeter, and volume formulas.
- ✓Practice mental math - multiplication tables, squares, and cubes.
- ✓Show your work - avoid careless errors by writing out steps.
- ✓Check your answers - substitute back to verify.