Lesson 2
Mathematics
In This Lesson
1. Algebra
Algebra forms the foundation of the USTET mathematics section.
Linear Equations
- Standard form: ax + b = c
- Slope-intercept form: y = mx + b (m = slope, b = y-intercept)
- Point-slope form: y - y₁ = m(x - x₁)
Example: Solve 3x + 7 = 22
3x = 22 - 7 = 15
x = 5
Quadratic Equations
- Standard form: ax² + bx + c = 0
- Quadratic formula: x = (-b ± √(b²-4ac)) / 2a
- Discriminant: b² - 4ac determines number of solutions
b² - 4ac > 0
Two real solutions
b² - 4ac = 0
One real solution
b² - 4ac < 0
No real solutions
Factoring Techniques
- • Common Factor: ax + ay = a(x + y)
- • Difference of Squares: a² - b² = (a + b)(a - b)
- • Perfect Square Trinomial: a² + 2ab + b² = (a + b)²
- • FOIL Method: (a + b)(c + d) = ac + ad + bc + bd
2. Geometry
Essential geometric formulas and concepts for the USTET.
2D Shapes - Area & Perimeter
Rectangle
Area = length × width
Perimeter = 2(l + w)
Triangle
Area = ½ × base × height
Perimeter = a + b + c
Circle
Area = πr²
Circumference = 2πr
Trapezoid
Area = ½(b₁ + b₂) × h
3D Shapes - Volume & Surface Area
Rectangular Prism
V = l × w × h
SA = 2(lw + lh + wh)
Cylinder
V = πr²h
SA = 2πr² + 2πrh
Sphere
V = (4/3)πr³
SA = 4πr²
Cone
V = (1/3)πr²h
Important Theorems
- Pythagorean Theorem: a² + b² = c² (right triangles)
- Triangle Angle Sum: Sum of interior angles = 180°
- Polygon Interior Angles: (n-2) × 180° where n = number of sides
- Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
- Midpoint Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
3. Statistics
Measures of central tendency and data analysis.
Measures of Central Tendency
Mean
Sum of values ÷ Count
(Average)
Median
Middle value when sorted
(Middle)
Mode
Most frequent value
(Most common)
Other Statistical Measures
- Range: Maximum value - Minimum value
- Variance: Average of squared deviations from mean
- Standard Deviation: √Variance (spread of data)
- Weighted Mean: Σ(value × weight) ÷ Σweights
Probability
- Basic Probability: P(A) = Favorable outcomes ÷ Total outcomes
- Complement: P(not A) = 1 - P(A)
- Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
- Multiplication Rule: P(A and B) = P(A) × P(B) (independent events)
4. Number Theory
Properties of numbers and divisibility rules.
Divisibility Rules
GCF and LCM
- GCF (Greatest Common Factor): Largest number that divides both numbers
- LCM (Least Common Multiple): Smallest number divisible by both numbers
- Relationship: GCF(a,b) × LCM(a,b) = a × b
5. Word Problems
Common types of word problems on the USTET.
Problem Types
- Age Problems: Set up equations based on current and future/past ages
- Distance Problems: Distance = Rate × Time (d = rt)
- Work Problems: Combined rate = 1/time₁ + 1/time₂
- Mixture Problems: Amount × Concentration = Total amount of solute
- Percentage Problems: Part = Percentage × Whole
Example: Distance Problem
Two cars start from the same point. Car A travels north at 60 km/h. Car B travels south at 40 km/h. How far apart are they after 2 hours?
Car A distance: 60 × 2 = 120 km
Car B distance: 40 × 2 = 80 km
Total distance apart: 120 + 80 = 200 km
6. Problem-Solving Strategies
Use POLYA's four-step method for solving math problems.
POLYA's Problem-Solving Method
- Understand the Problem
What is given? What is asked? Draw a diagram if helpful.
- Devise a Plan
Find a pattern, make an equation, work backwards, or guess and check.
- Carry Out the Plan
Execute your strategy step by step. Show your work.
- Look Back and Check
Verify your answer. Does it make sense? Is it reasonable?
USTET Math Tips
- ✓Memorize formulas - know them by heart for quick recall.
- ✓Practice mental math - speed is important in a timed test.
- ✓Show your work - write out steps to avoid careless errors.
- ✓Check answers - substitute back to verify your solution.